Mathematics - Drs International School

Transcription

INTERNATIONAL BACCALAUREATE ORGANIZATION
Primary Years Programme
Mathematics: Scope and Sequence
Primary Years Programme
Mathematics: Scope and Sequence
Draft January2003
© International Baccalaureate Organization 2003
Previously published as draft in 1998
Organisation du Baccalauréat International
Route des Morillons 15
1218 Grand-Saconnex
Genèva,
SWITZERLAND
Planning mathematical inquiry
To plan authentic mathematical inquiry, teachers must consider the following questions.
• What do we want students to learn?
• What do teachers need to learn about this topic?
• How best will students learn?
• How will we know what students have learned?
What do we want students to learn?
The mathematics scope and sequence framework identifies the expectations considered essential in the
Primary Years Programme (PYP). These expectations are arranged in five interwoven strands of
knowledge. In number and pattern and function, students inquire into our number system, and its
operations, patterns and functions. This is where students become fluent users of the language of
arithmetic, as they learn to understand its meanings, symbols and conventions. These two strands are
best taught as stand-alone mathematical topics, although aspects can be successfully included in the
units of inquiry.
The remaining strands, data handling, measurement and shape and space, are the areas of
mathematics that other disciplines use to research, describe, represent and understand aspects of their
domain. Mathematics provides the models, systems and processes for handling data, making and
comparing measurements, and solving spatial problems. As a discipline it has no interest in, or ability
to determine, the content of charts, graphs, tables, shapes or measurements. The content must come
from the theme or area under study. Consequently, topics in these three strands are best studied in
authentic contexts such as the transdisciplinary units of inquiry.
What do teachers need to learn about this topic?
The PYP views mathematics not as a fixed body of knowledge to be transmitted, but as a way of
thinking and a language for understanding meaning. To study mathematics is to inquire into this
language and to learn to think in this way. We believe teachers can use the eight key concepts and
related questions (Figures 5 and 6 in Making the PYP happen) to guide their own inquiry. By engaging
in inquiry themselves, teachers will not only achieve a deeper understanding of the mathematical
issues involved, but will also provide a model for their students of the teacher as a learner.
Sample questions for each expectation and for each strand have been provided. Each of the questions
may be linked to one or more concepts by the teacher. Some of the sample questions have been linked
to an appropriate concept as examples.
Personal knowledge of the subject matter is of key importance. What teachers understand themselves
will shape how well they select from activities and texts available, and how effectively they teach. The
teacher’s personal interest in, and development of, the discipline should be maintained through regular
professional development, reading professional journals and regular contact with colleagues who share
their commitment to teaching mathematics through inquiry.
PYP curriculum documents: mathematics © IBO 2003
3.1
How best will students learn?
Cognitive psychologists have described the stages through which children learn mathematics. It is
useful to identify these stages when designing developmentally appropriate learning experiences. The
scope and sequence document uses and identifies these stages as follows.
•
Constructing meaning
Teachers plan activities, through which students construct meaning from direct experience, by
using manipulatives and conversation.
•
Transferring meaning into signs and symbols
Teachers connect the notation system with the concrete objects and associated mathematical
processes. The teacher provides the symbols for the students. Students begin to describe their
understanding using signs and symbols.
•
Understanding and applying
Teachers plan authentic activities in which students independently select and use appropriate
symbolic notation to process and record their thinking.
As they work through these stages, students use certain processes of mathematical reasoning.
•
•
•
•
They use patterns and relationships to analyse the problem situations upon which they are working.
They make and evaluate their own and each other’s ideas.
They use models, facts, properties and relationships to explain their thinking.
They justify their answers and the processes by which they arrive at solutions.
In this way, students validate the meaning they construct from their experiences with mathematical
situations. By explaining their ideas, theories and results, both orally and in writing, they invite
constructive feedback and also lay out alternative models of thinking for the class. Consequently, all
benefit from this interactive process.
What does a PYP mathematics classroom look like?
In a PYP classroom, mathematics is a vital and engaging part of students’ lives. Students in the
classroom are very active with an underlying sense of organization and cooperation. Teachers and
students are asking questions of each other, trying out and demonstrating ideas in small and large
groups, using the language of mathematics to describe their thinking, generating data to look for
patterns, and making conjectures.
The value of open questions cannot be over-emphasized. When asked a closed question, students
quickly learn that the accuracy of their answer is important. This may develop into an anxiety over
getting the answers “wrong”. Open questions help students to develop their understanding and allow
them an opportunity to think about their work and to verbalize their learning. Such probing questions
quickly reveal those who have understood and those who have guessed. From the earliest stages,
correct and appropriate mathematical language should be used. Consistent use will enable students to
remember and use relevant vocabulary with confidence.
The use of appropriate technology in mathematics influences and enhances student learning. The
computer area is well stocked with innovative software that encourages the application of mathematics
skills and problem solving.
A wide variety of materials is available to all, ranging from student collections of keys, foreign money
and seashells to store-bought cubes and counters, pattern blocks and geoboards. Everyday tools, such
as measuring jugs, calculators and even cereal boxes, are in use. It is important that students are given
the opportunity to select from a range of materials to enable them to generalize their understanding.
3.2
The constant use of appropriate manipulatives is an essential aspect of PYP mathematics. A variety
of different materials should be used for each strand and expectation, and should continue to be used
all the way through the programme.
The number of mathematics resources available is impressive. There are lists, tables and charts on
display, showing written and numerical data, about which relevant questions are asked and answered.
Colourful and thought-provoking posters and children’s work cover the walls. On the bookshelves are
many resource books for the teacher and students, including a wide variety of textbooks, mathematical
dictionaries and encyclopedias as well as children’s literature that focuses on mathematical ideas. The
general supplies area has a variety of paper for recording mathematical ideas: different sized squared
paper, dot paper and scribble paper. There may be a video area where high-quality video tapes show
how mathematics is used outside the classroom. It is clear where things belong and how they are used.
The mathematics classroom does not work on its own. The students may visit younger children to help
them with their investigations, or work with older children to travel about town in small groups on a
treasure hunt, asking and answering questions about dates, times, distance, prices and more. Other
teachers and parents come to share their interests and expertise. The school nurse and school
administrators are involved in providing information and participating in surveys. The school’s parking
lot and local shops are visited when collecting data. The community at large provides innumerable
opportunities for budding mathematicians to practise their craft. Wherever possible, the mathematical
ideas students are experiencing should be supported at home and given real-life significance, for
example, cooking; buying clothes or toys and comparing prices; and calculating materials required for
decorating.
How does a PYP mathematics classroom work?
Inquiry-based units of study are entry points into mathematics learning through which students will
experience what it is like to think and act as mathematicians. Students and teachers together identify
things they already know that might be relevant to an inquiry, what they want to know, what they need
to know to answer their questions, and how best they might find that out. They are encouraged to use
multiple strategies, developing an understanding of which strategies are most effective and efficient.
The students must be given an opportunity to communicate their mathematical thinking and strategies
to others and to have time to reflect upon them.
The teacher spends time in different ways: walking around while students are working alone, in pairs,
in small groups or even as a whole class; asking key questions; challenging the students’ thinking and
prompting them to take ideas a step further; and jotting down notes to inform the next stages of
learning. The teacher might also gather together a group of students, with a particular interest or
problem, to provide more specific help through guidance and practising together.
One of the most important aspects of the teacher’s role is to encourage appropriate mathematical
discussion among the student mathematicians, demonstrating the nature of mathematical discourse and
the development of conjectures. Students follow simple, polite rules when talking to each other, build
on previously mentioned ideas, support others in the various stages of learning and share their
discoveries in a congenial atmosphere. Students see writing down their ideas as a natural step in the
process of communicating important ideas. They record their ideas in a variety of ways, including
drawing pictures, recording numbers and writing in mathematics journals or log books.
An exemplary PYP mathematics classroom consists of a very active and busy community of learners,
with the teacher constantly finding ways to combine the students’ needs and interests with the goals of
the curriculum, using engaging and relevant tasks.
Why is a PYP mathematics classroom the way it is?
Mathematics is a language that is used to describe the natural world. Students need to build an
understanding of previously made discoveries, as well as to discover and describe their own
mathematical ideas.
3.3
Traditionally, mathematical knowledge has been disassembled in schools; broken into unrelated skillsbased activities. Our vastly expanded knowledge base about learning mathematics, however, tells us
that people assemble, or construct, mathematical knowledge. This requires us to look at mathematics
not as a fixed body of knowledge to be transmitted, but as a language and a way of thinking. It is our
task as teachers to facilitate this process.
Mathematics, like language, can be seen as a service discipline to other parts of the curriculum, providing
tools, symbolic language and ways of thinking to the scientist, social scientist and musician; but
mathematics is a fascinating discipline in its own right. It is the joy and satisfaction of solving problems and
finding patterns that has captured and stimulated the most creative minds throughout the ages.
How will we know what students have learned?
Appropriate and regular assessment is essential in monitoring what students have learned. There
should be ongoing formative assessments as well as summative assessments. Assessment activities
should be carefully planned, and opportunities for students to self-assess, using different methods,
should be included.
Examples of assessments are included in the scope and sequence document, along with activities and
key questions for each expectation. English as additional language (EAL) learners may require special
attention in order to fully participate in the activities. When the mathematical understandings of EAL
learners are being assessed, care must be taken not to underestimate the students’ ability due to their
lack of language proficiency.
The age bands of the scope and sequence document should be viewed only as a guide. A student’s
ability to successfully understand and apply mathematical knowledge depends less on their age and
more on the experience and support they receive. Some common difficulties have been described in
the teachers’ notes where appropriate.
Record keeping should be simple and readily accessible for the teacher and the student. Examples of
significant progress or development in mathematical understanding should be included in the student
portfolio.
* See glossary for explanation of italicized terms.
Recommended resources
Duncan A. 1996. What Primary Teachers Should Know About Maths. Hodder & Stoughton.
(ISBN 0-304-64377-3). This comprehensively informative book follows an inquiry-based format.
Baratta-Lorton M. 1995. Mathematics Their Way. Addison Wesley. (ISBN 0-201-86150-X). This is an
excellent resource for data handling and graphing activities for 3–7 year olds.
National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics.
(ISBN 0873534808). Web site: www.nctm.org.
3.4
Glossary of PYP mathematical terms
acute angle
An angle of less than 90°.
addend
Any number added in an addition.
addition
Putting two or more sets together to create a new set. The inverse operation of
subtraction. Plus, and, increase by, more than, count on, find the total, what is the sum
of? can all be used to mean addition.
algebra
A mathematical language where relationships can be expressed in numbers, symbols
or letters.
algebraic formula
A sequence of symbols used to express a rule or result.
algorithm
A method for calculating solutions.
angle
An amount of rotation, measured in degrees.
approximation
An estimation of a quantity that is close to the actual value but not totally accurate.
area
The amount of surface an object covers, usually measured in square units.
array
An arrangement showing quantities in rows and columns. Multiplication can be shown
in this way to demonstrate square and rectangular numbers.
column
↓
■■■■
row → ■ ■ ■ ■
■■■■
associative property
3 % 4 = 12
When the order in which mathematical operations are carried out does not affect the
result. This applies to addition and multiplication, for example:
(2 + 3) + 4 = 2 + (3 + 4) = 9, (5 % 6) % 7 = 5 % (6 % 7) = 210
axis (plural axes)
A line on a graph, usually either horizontal or vertical.
axis of symmetry
An imaginary line around which shapes are symmetrical. It can be a fold line in a 2-D
shape or an axis around which shapes turn in rotational symmetry.
bar graph
A graph in which data is represented by horizontal or vertical bars.
base 10 system
A number system based on grouping in tens.
3.5
bearing
The position of something or direction of movement, relative to a fixed point. It is
usually measured in degrees between the direction of travel and magnetic north,
expressed as three figures.
N
040°
It can also be expressed from north or south in the east or west direction. The angle in
this case will always be less than 90°.
S 80° W
S
block graph
A graph in which data is shown by blocks, representing one unit, arranged in columns.
capacity
The space within a container, often referred to as volume.
cardinal number
A number denoting quantity (2, 3, 4) as opposed to order (1st, 2nd). See ordinal number.
Carroll diagram
A way of sorting any yes/no information using labelled boxes.
carrying
less than 6
not less than 6
1, 3, 5
7, 9
2, 4
6, 8, 10
odd
not odd
The process of marking an extra number to a column when adding or multiplying
multiple digit numbers.
568
+ 354
922
11
Cartesian coordinates
3.6
Cartesian coordinates describe the position of a point on a graph or chart, so the lines
of the axes must be numbered and not the spaces, also called a grid reference. See
coordinates.
chord
A straight line, joining two points on the circumference of a circle, that does not pass
through the centre of the circle.
circle
A round, plane shape whose boundary (circumference) is equidistant from its centre.
circle graph
See pie chart.
circumference
The distance around the outside of a circle, which can be calculated by multiplying the
diameter by (d).
commutative property
When the order in which numbers are arranged does not affect the result. This applies
to addition and multiplication, for example:
4 + 3 + 7 = 7 + 4 + 3 = 14, 5 % 3 % 4 = 4 % 3 % 5 = 60
complementary angles
Two angles whose sum is 90°.
congruency
The property of having the same size and shape.
congruent angles
Two angles of the same size.
conservation
The property of remaining the same despite changes such as physical arrangement.
coordinates
A pair of numbers (or a letter and number) placed in order to describe position. The
first number describes the position on a horizontal axis, the second number describes
the position on a vertical axis.
corner
Where two or more edges meet.
Cuisenaire rods
Commercially manufactured coloured rods of different sizes used to develop a sense
of number facts and an understanding of fractions.
database
An organized and structured set of information, often held on a computer.
decimal notation
A way of showing values of less than 1 using place value and a dot to separate the
whole units and fractional parts of a unit.
decomposition
See regrouping.
degree
1. A unit of measure for angles.
2. A unit of measure for temperature (degrees centigrade, °C, used in the metric system).
denominator
The number below the line in a fraction, showing the number of equal parts in the
whole.
¾ ←
denominator
diagonal
A straight line passing through a shape and joining any two non-adjacent corners.
diameter
A line passing through the centre of a circle, with endpoints on the circumference.
difference
The result of a subtraction calculation.
digit (figure)
A symbol used to write numbers.
3.7
dividend
The amount to be divided in a division calculation.
division
Breaking down a set into smaller sets of the same size as each other. The inverse
operation of multiplication. The dividend is the amount to be divided, the divisor is the
number of sets (or number in each set), and the quotient is the answer.
45 + 9 = 5
↑
dividend
↑
divisor
↑
quotient
divisor
Number of equal sets or groups into which the dividend is divided.
EAL
English as an additional language. Students learning in a language other than their
mother tongue.
edge
Where the two faces of a shape meet. It can be curved or straight.
equation
A mathematical sentence balanced by an equals (=) sign. Equations can be numerical
or algebraic.
equilateral triangle
A triangle that has three equal sides and equal internal angles.
equivalence
Expressing the same value in different ways, for example, 0.75 = ¾ = 6/8 = 75%.
estimate
Arriving at a rough answer by making suitable approximations.
experimental
probability
When considering the likelihood of an event, it is what actually happens (how many
times a six is thrown in 100 throws of a dice). See theoretical probability.
exponential notation
A quick way of writing repeated multiplication, for example, 6³, where 3 is the
exponent, indicates that 6 is to be multiplied by itself 3 times (6 % 6 % 6).
6³ (6 to the power of 3) is in exponential notation. When the exponent is 2, the number
is squared (squared number). When the exponent is 3, the number is cubed (cubed
number).
face
A surface of a 3-D shape.
factor
A number that divides into another number exactly, for example, 8 is a factor of 8, 16,
24, 32 etc. The number 1 is a factor of all whole numbers.
Fibonacci sequence
An infinite sequence in which each number is the sum of the two preceding numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34…
formula
A set pattern for calculating a variable. It is usually written in algebraic form.
For example, a formula to find the volume of a box is length % width % height so
V = LWH.
fraction
A part of a whole. It can also be expressed as a percentage or a decimal.
function
A relationship that should be applied to all numbers in a particular pattern, for example,
add 2, divide by 3, double then subtract 1.
3.8
geoboard
A commercially manufactured plastic pinboard, used with elastic bands to make and
explore shapes.
graph
A pictorial representation of information.
improper fraction
A fraction in which the numerator is greater than the denominator, for example, 7/4 , 9/2.
integer
Any whole number, including those below zero.
inverse operations
Opposite operations, for example, addition and subtraction, multiplication and division.
isosceles triangle
A triangle that has two sides of equal length.
isometric dot paper
Paper with equally spaced dots.
kinaesthesia
An awareness of the position and movement of the parts of the body by means of
sensory organs.
kite
A quadrilateral with two pairs of equal adjacent sides, perpendicular diagonals, and one
line of symmetry. If one of the internal angles is reflex, the kite becomes an arrowhead.
line
The shortest distance between two points.
line graph
A graph that is used to represent the relationship between two continuous variables
(for example, time and temperature). The line may be straight or curved.
line of symmetry
A line that divides a shape in half so that each side is a mirror image of the other.
manipulative
A concrete material.
mass
The amount of matter in an object. The term weight is often incorrectly used for mass.
mean
The arithmetic mean is a way of measuring the average in a set of data. It is calculated
by finding the sum of the data and dividing by the number of sets of data, for example,
in the data set 7, 9, 13, 18, 23 the mean is 14 (7 + 9 + 13 + 18 + 23 = 70; 70 ÷ 5 = 14).
Commonly referred to as the average.
median
The middle number in a set of data, for example, in the data set 7, 9, 13, 18, 23 the median
is 13. In the data set 7, 9, 13, 15, 18, 23 the median is 14 (13 + 15 = 28; 28 + 2 = 14).
mixed number
A whole number and a fraction, for example, 3⅛.
mode
The number that occurs most frequently in a set of data, for example, in the data set
3, 5, 6, 5, 7, 5, 8, 3, 5 the mode is 5.
model
To carry out the activity in a practical way using manipulatives. To demonstrate or
show a method.
multiple
A natural number that can be divided by another, a certain number of times, without a
remainder, for example, 12 is a multiple of 2, 3, 4, 6 and 12.
3.9
multiplication
A quick method of adding together groups of the same cardinality (repeated addition).
The inverse function of division. The numbers to be multiplied are factors and the
result is the product.
5 % 9 = 45
↑
↑
↑
factor factor
product
natural number
Any of the counting numbers (1, 2, 3, 4…).
negative number
Any number below zero.
net
A 2-D shape that can be folded to create a 3-D shape.
non-standard units
of measurement
(arbitrary units)
Any item used in the development of the concept of measurement, for example, feet,
hand spans, paper clips or cubes.
number facts
Basic addition, subtraction, multiplication and division facts that students learn and
use automatically.
numerator
The number above the line in a fraction, showing the number of parts being referred
to.
numerator
→ ¾
numerical probability
scale
The chance of an event happening is expressed between 0 and 1, or 0% and 100%,
where 0 is impossible. Also, the term 50:50 expresses an even chance.
obtuse angle
An angle of more than 90° and less than 180°.
one-to-one (1–1)
correspondence
Matching a number with an object. This can be by using numerals or simply by
matching items to items (4 paper hats for 4 teddy bears).
ordinal number
A number that shows the relative position or place in a sequence: 1st, 2nd, 3rd.
parallel lines
Lines that always stay the same distance apart and never meet.
parallelogram
A quadrilateral with opposite sides that are parallel and equal in length.
pattern
An arrangement of objects, shapes or numbers that repeats or changes in a regular way.
percentage
A way of expressing a fraction of an amount using a denominator of 100.
perimeter
The total distance around the outside of a 2-D shape.
perpendicular lines
Lines that cross at right angles.
pictograph
Data that is represented by pictures or symbols.
pie chart
A chart or graph in which a circle represents the whole and is then divided to represent
the data.
3.10
place value
A method of recording numbers, using the digits in different positions to represent
different values.
plane
A flat surface that can be vertical, horizontal or sloping.
polygon
A 2-D shape with many sides.
polyhedron
(plural polyhedra)
A 3-D shape with four or more faces.
prime number
Any number that has only two factors, 1 and itself. For example, 2, 3, 5, 7, 11…
The number 1 is not a prime number as it only has one factor.
product
The answer to a multiplication calculation.
quadrants
The quarters of a grid. The first quadrant refers to the positive quarter of the grid.
4
First quadrant
3
2
1
-4
-3
-2
-1
1
2
3
4
-2
-3
-4
quadrilateral
A polygon with four sides. They can be classified by their angles or their sides.
quotient
The answer to a division calculation.
radius
A line that has its endpoints in the centre of a circle and on the circumference. It is
half the length of the diameter.
range
The spread of numbers in a set of data (the difference between the highest and lowest
values). The range of numbers in the set of data 7, 9, 13, 18, 23 is 16 (23 − 7).
ratio
The relationship between two numbers or quantities.
3.11
ray
A part of a line with only one endpoint.
real-life graph
A 3-D graph where each unit is represented by a single object.
rectangle
A parallelogram with four right angles (angles of 90°).
reflective symmetry
The exact matching of parts on either side of a straight line. It is sometimes called line
symmetry, mirror symmetry or bilateral symmetry.
reflex angle
An angle of more than 180°.
regrouping
The exchange and rearrangement of tens for units, or hundreds for tens, in order to
calculate multiple digit subtraction equations, for example:
16712
59
113
The 7 tens have been “regrouped” into 6 tens and 10 units to allow the subtraction to
take place. This method should only be used when a full understanding of place value
has been achieved.
rhombus
A quadrilateral with two pairs of parallel sides, all sides of equal length, and opposite
angles that are equal.
right-angled triangle
A triangle that has one angle of 90°.
roots
Equal factors of a product. Square root is two equal factors, cube root is three equal
factors, for example, the square root of 81 is 9, √81 = 9 ( 9 % 9 = 81). The cube root of
27 is 3 ³√27 = 3 (3 % 3 % 3 = 27).
rotational symmetry
When a shape fits onto itself after a turn of less than 360°, it is said to have rotational
symmetry. The number of turns required to return to its original position describe the
order of symmetry. For example, a square has a rotational symmetry of order 4.
scale
The relative size or extent of something, a ratio of size.
scalene triangle
A triangle that has three sides of different lengths.
sector
A slice of a circle.
segment
A part of a circle bound by a chord and the circumference.
set, subset, superset
A way of grouping and classifying numbers or objects, for example, a set may be cats,
a subset would be all black cats and a superset would be all pets.
side
One of the lines forming the boundary of a plane shape.
spreadsheet
A type of database where information is set out in a table.
square
A rectangle with sides of equal length.
standard units of
measurement
Internationally recognized units of measurement, for example, centimetres, inches,
kilograms, seconds, minutes.
3.12
straight angle
An angle of 180°.
subtraction
The inverse operation of addition. It calculates the difference between two amounts.
Minus, take away, decrease by, less than, fewer than, the difference between, count
back can also be used to mean subtract.
sum
The result of adding numbers together, also referred to as the total.
supplementary angles
Two angles whose sum is 180°.
surface area
The sum of the area of the faces of a 3-D shape.
survey
A method of obtaining information, usually by asking people questions.
symmetry
Having exactly matching parts facing each other around a line or axis.
t-chart
A chart that displays information before and after a function has been applied.
In
Out
2
3
4
5
6
7
8
9
(the function is + 4)
tessellation
A tiling pattern where shapes fit together without overlapping or gaps.
theoretical probability
When considering the likelihood of an event, it is what is expected to happen (how
many times 6 might be thrown in 100 throws of a dice). See experimental probability.
three-dimensional
(3-D) shape
A solid shape that can be measured in three dimensions – length, width and height.
Examples include sphere, cylinder, cone, prism, pyramid and poyhedron.
total
The result of adding numbers together, also referred to as the sum.
trapezium (UK)/
trapezoid (US)
A quadrilateral with one pair of parallel sides.
tree diagram
A branching diagram to show all possible outcomes.
triangle
A plane shape with three straight sides.
turn symmetry
See rotational symmetry.
two-dimensional (2-D)
shape
A flat shape that can be measured in two dimensions (length and width). Examples
include circle, square, triangle and polygon.
3.13
Venn diagram
A way of displaying data sorted by characteristics. Where the items may belong in two
of the groupings, the groupings overlap.
Flat and curved faces
Flat faces
Cube
Cuboid
Pyramid
Cone
Cylinder
Sphere
Ovoid
Curved faces
vertex (plural vertices)
A point where lines or edges meet.
volume
The amount of space a solid shape occupies. It is usually measured in cubic units.
weight
The measure of the heaviness of an object. The term is often incorrectly used for mass.
(It is calculated by multiplying mass by gravity, which explains why astronauts are
“weightless” even though their mass remains the same.)
whole number
Any of the natural numbers, together with zero (0, 1, 2, 3, 4…).
3.14
Mathematics scope and sequence overview
In addition to the following strands, students will have the opportunity to identify and reflect upon “big ideas” by making connections between the questions asked and the concepts that drive the inquiry.
They will become aware of the relevance these concepts have to all of their learning.
Strand
By the end of this age range,
children aged 3–5 will:
students aged 5–7 will:
Data handling • sort and label real objects into sets by attributes
• sort and label objects into sets by one or more attribute
• discuss, compare and create sets from data that has subsets • design a survey and systematically collect, organize and
using tree, Carroll, Venn and other diagrams
record the data in displays: pictograph, bar graph, circle
• create a graph of real objects and compare quantities using • discuss and compare data represented in teacher-generated
graph (pie chart), line graph
number words
diagrams: tree, Carroll, Venn
• design a survey, process and interpret the data
•
create, interpret, discuss and compare data displays
• discuss and identify outcomes that will happen, won’t
• collect, display and interpret data for the purpose of finding • collect and display data in a bar graph and interpret results
(pictograph, pie chart, bar/line graph) including how well
happen and might happen.
information
• use the scale on the vertical axis of a bar graph to represent
they communicate information
• understand the purpose of graphing data
large quantities
• create a pictograph and simple bar graph from a graph of • find, describe and explain the mode in a set of data and its • find, describe and explain the range, mode, median and
real objects, and interpret data by comparing quantities:
more, fewer, less than, greater than
• discuss, identify, predict and place outcomes in order of
likelihood: impossible, unlikely, likely and certain.
Measurement • identify, compare and describe attributes of real objects
• estimate, measure, label and compare using non-standard
and situations: longer, shorter, heavier, empty, full, hotter,
units of measurement: length, mass, time and temperature
colder
• understand why we use standard units of measurement to
measure
• identify, compare and sequence events in their daily
routine: before, after, bedtime, storytime, today, tomorrow. • use a calendar to determine the date, and to identify and
sequence days of the week and months of the year
• estimate, identify and compare lengths of time: second,
minute, hour, day, week, month
• read and write the time to the hour, half hour and quarter
hour.
mean in a set of data and understand their use
use
• create and manipulate an electronic database for their own
• understand the purpose of a database by manipulating the
purposes
data to answer questions and solve problems
•
set up a spreadsheet, using simple formulas, to manipulate
• use probability to determine mathematically fair and unfair
data and to create graphs
games and to explain possible outcomes.
• use a numerical probability scale 0 to 1 or 0% to 100%
• determine the theoretical probability of an event and explain
why it might differ from experimental probability.
• estimate, measure, label and compare using formal
•
•
•
•
•
methods and standard units of measurement: length, mass,
time and temperature
select appropriate tools and units of measurement
describe measures that fall between numbers on a measure
scale: 3½kg, between 4cm and 5cm
estimate, measure, label and compare perimeter and area
model addition and subtraction using money
read and write the time to the minute and second, using
intervals of 10 minutes, 5 minutes and 1 minute, on 12-hour
and 24-hour clocks.
• select and use appropriate standard units of measurement
•
•
•
•
•
•
•
•
•
•
Shape and
space
• sort, describe and compare 3-D shapes according to
• use what they know about 3-D shapes to see and describe
attributes such as size or form
2-D shapes
• explore and describe the paths, regions and boundaries of • sort and label 2-D and 3-D shapes using appropriate
their immediate environment (inside, outside, above, below)
mathematical vocabulary: sides, corners, circle, sphere,
and their position (next to, behind, in front of, up, down).
square, cube
• create 2-D shapes
• find and explain symmetry in their immediate environment
• create and explain simple symmetrical designs
• give and follow simple directions, describing paths, regions
and boundaries of their immediate environment and their
position: left, right, forward and backward.
• sort, describe and model regular and irregular polygons:
•
•
•
•
•
•
triangles, hexagons, trapeziums
identify, describe and model congruency in 2-D shapes
combine and transform 2-D shapes to make another shape
create symmetrical patterns, including tessellation
identify lines and axes of reflective and rotational
symmetry
understand an angle as a measure of rotation by comparing
and describing rotations: whole turn; half turn; quarter turn;
north, south, east and west on a compass
locate features on a grid using coordinates.
• use the geometric vocabulary of 2-D and 3-D shapes:
parallel, edge, vertex
• classify, sort and label all types of triangles and quadrilaterals:
•
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•
when estimating, describing, comparing and measuring
use measuring tools, with simple scales, accurately
understand that the accuracy of a measurement depends on
the situation and the precision of the tools
develop procedures for finding area, perimeter and volume
determine the relationships between area, perimeter and volume
estimate, measure, label and compare, using formal
methods and standard units of measurement, the
dimensions of area, perimeter and volume
use decimal notation in measurement: 3.2cm, 1.47kg
understand that an angle is a measure of rotation
measure and construct angles in degrees using a protractor
use and construct timetables (12-hour and 24-hour) and
time lines
determine times worldwide.
scalene, isosceles, equilateral, right-angled, rhombus,
trapezium, parallelogram, kite, square, rectangle
understand and use the vocabulary of types of angle:
obtuse, acute, straight, reflex
understand and use geometric vocabulary for circles:
diameter, radius, circumference
use a pair of compasses
understand and use the vocabulary of lines, rays and
segments: parallel, perpendicular
describe, classify and model 3-D shapes
turn a 2-D net into a 3-D shape and vice versa
find and use scale (ratios) to enlarge and reduce shapes
use the language and notation of bearing to describe position
read and plot coordinates in four quadrants.
3.15
Strand
Pattern and
function
children aged 3–5 will:
• find and describe simple patterns
• create simple patterns using real objects.
• create, describe and extend patterns
• analyse patterns in number systems to 100
• recognize, describe and extend patterns in numbers: odd and • recognize, describe and extend more complex patterns in
even, skip counting, 2s, 5s and 10s
• identify patterns and rules for addition: 4 + 3 = 7,
3 + 4 = 7 (commutative property)
• identify patterns and rules for subtraction:
7 – 3 = 4, 7 – 4 = 3
• model, with manipulatives, the relationship between
addition and subtraction: 3 + 4 = 7, 7 – 3 = 4.
•
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•
•
•
•
•
Number
•
•
•
•
•
•
•
•
•
read, write and model numbers to 20
count, compare and order numbers to 20
estimate quantities to 10
use ordinal numbers to describe the position of things in
a sequence
model number relationships to 10: “Show me one more
than three, take two away from these cubes”
use the language of mathematics: more, less, number
names, total
use 1–1 correspondence
explore the conservation of number through the use of
manipulatives
select and explain an appropriate method for solving a
problem.
• read, write, and model numbers, using the base 10 system, • read, write and model numbers, using the base 10 system,
•
•
•
•
•
•
•
•
•
•
to 100
count (in 1s, 2s, 5s and 10s), compare and order numbers
to 100
use mathematical vocabulary and symbols of addition and
subtraction: add, subtract, difference, sum, +, –
read, write and model addition and subtraction to 20 (with
and without regrouping)
automatically recall addition and subtraction facts to 10
describe the meaning and use of addition and subtraction
explore and model multiplication and division using their
own language/methods
use fraction names (half, quarter) to describe part and
whole relationships
estimate the reasonableness of answers
problem.
•
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•
•
•
•
•
•
•
•
•
3.16
numbers
understand and use the relationship between addition and
subtraction: 4 + 3 = 7, 7 – 3 = 4
identify patterns and rules for multiplication and division:
4 × 3 = 12, 3 × 4 = 12, 12 + 3 = 4, 12 + 4 = 3
model, with manipulatives, the relationship between
multiplication and division
multiplication and addition (repeated addition)
division and subtraction
model multiplication as an array
understand and use number patterns to solve problems
(missing numbers).
to 1000
count, compare and order numbers to 1000
count in 3s, 4s, 6s, and explore other numbers
use number patterns to learn multiplication tables: 1s, 2s,
4s, 5s, 10s
automatically recall basic addition and subtraction facts
model addition and subtraction equations to 1000 (with
and without regrouping)
use mathematical vocabulary and symbols of multiplication
and division: times, divide, product, quotient, %, +
use and describe multiple strategies to solve addition,
subtraction, multiplication and division problems
read, write and model multiplication and division problems
compare fractions using manipulatives and using fractional
notation
model addition and subtraction of fractions with the same
denominator
use mathematical vocabulary and symbols of fractions:
numerator, denominator, equivalence
understand and model the concept of equivalence to 1:
two halves = 1, three thirds = 1
reasonably estimate answers: rounding and approximation
problem.
• understand and use the relationship between multiplication
and addition
• understand and use the relationship between multiplication
•
•
•
•
•
•
and division (inverse function)
understand and use the relationship between division and
subtraction
model and explain number patterns
use real-life problems to create a number pattern, following
a rule
develop, explain and model simple algebraic formulas in
more complex equations: x + 1 = y, where y is any even
whole number
model exponents as repeated multiplication
understand and use exponents and roots as inverse
functions: 9², √81.
• read, write and model numbers, using the base 10 system,
to millions and beyond; and to thousandths and beyond
• automatically recall and use basic number facts
• create and solve multiple digit multiplication and division
•
•
•
•
•
•
•
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•
•
•
•
•
•
problems
read, write and model addition and subtraction of fractions
with related denominators
read, write and model improper fractions and mixed
numbers
compare and order fractions
model equivalency of fractions: 2/4 = ½
simplify fractions
use the mathematical vocabulary of fractions: improper,
mixed numbers
read, write and model the addition and subtraction of
decimals to the thousandths
read, write and model multiplication and division of
decimals (with reference to money)
round decimals to a given place or whole number
read, write and model percentages
interchange fractions, percentages and decimals
find and use ratios
read, write and model integers
read, write and model addition and subtraction of integers
(negative numbers)
read, write and model exponential notation
select and defend the most appropriate and efficient method
of solving a problem: mental estimation, mental arithmetic,
pencil and paper algorithm, calculator.
Subject: mathematics
Age range: 3–5 years
Page 1 of 6
Overall expectations
Children will have the opportunity to identify and reflect upon “big ideas” by making connections between the questions asked and the concepts that drive the inquiry. They will become aware
of the relevance these concepts have to all of their learning.
Data handling: statistics and probability
Children will sort real objects by attributes, create graphs using real objects and compare quantities. They will discuss and identify outcomes that will happen, won’t happen or might happen.
Measurement
Children will identify and compare attributes of real objects, and events in their realm of experience.
Shape and space
Children will sort, describe and compare 3-D shapes and explore the paths, regions and boundaries of their immediate environment and their position.
Pattern and function
Children will find, describe and create simple patterns in their world.
Number
Children will read, write, count, compare and order numbers to 20. They will model number relationships to 10, develop a sense of 1–1 correspondence and conservation of number. They will
select and explain an appropriate method for solving a problem.
Content
Data handling: statistics
Data can be recorded,
organized, displayed and
understood in a variety of
ways to highlight similarities,
differences and trends.
What do we want children to learn?
How best will children
learn?
How will we know what
children have learned?
Notes for teachers
Specific expectations
Sample questions
Sample activities
Sample assessments
Resources and comments
The specific expectations must be
carefully addressed to ensure
learners have FULLY understood
a mathematical concept before
moving on. These expectations
represent the desired
understanding to be achieved by
the end of this age range.
Questions that address the key
concepts (Fig 5 Making the PYP
happen) challenge learners and
promote genuine understanding.
All of the sample questions can
be linked to a key concept. Some
examples are noted below in
bold.
All activities encompass some, or
many, of the specific
expectations and
transdisciplinary skills
(Fig 14 Making the PYP happen).
Assessments should be directly
related to the specific
expectations. Children should be
given the opportunity to
demonstrate their understanding
in a variety of ways.
Teachers should find ways to
ensure EAL learners understand
tasks and expectations.
Manipulatives facilitate
understanding for all learners.
Children will:
• sort and label real objects
into sets by attributes
What things belong together?
Why do they belong together?
connection
Can we sort them in another
way?
• create a graph of real
objects and compare
quantities using number
words.
What different ways are there
to show what we have done?
Which has more?
Which has the most?
Which has the least?
Can we see any patterns in
the graph? form
How can we describe the
information in these graphs?
form
How can we use number
words to describe what we see
on the graph?
Sorting
Children sort collections of
keys, nuts, bottle tops, blocks
and themselves to determine
attributes.
Children physically group
themselves by eye colour or
shoe colour.
Creating graphs
Every day, children
experience events that have
possibilities for graphing:
Who is present and who is
absent? What colour shoes are
we wearing? What are our
favourite characters in the
stories we have read?
Children create a real graph
using the wrappers from their
snacks glued on to a large
sheet of paper.
Children are presented with a
set of objects and can sort
them in two different ways
using attributes they have
identified. They can justify
their criteria.
Children are given a graph
and, when asked, can make
two true statements about it.
At this stage of development,
children should be asked to
compare and quantify the
results of the graph through
teacher-generated questions:
How many more? How many
fewer? Which is the most
numerous?
Children can make a graph of
the real objects provided.
They can describe the data
displayed using number
words and comparisons.
The teacher may record
results to demonstrate the use
of mathematical diagrams/
graphs in noting observations
and interpreting data. It is
important to remember that
the chosen format should
illustrate the information
without bias.
The units of inquiry will also
be rich in opportunities for
graphing.
3.17
Content
Data handling: probability
Appropriate qualitative
vocabulary may be introduced
such as “will happen”, “won’t
happen” and “might happen”,
as guided by the exploration
and the experiences of the
children.
Page 2 of 6
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Children can identify which
of these things will happen,
won’t happen or might
happen and they can suggest a
reasonable explanation why.
This idea is difficult for very
young children because they
view the world as a place of
all possibilities. The teacher
should try to introduce
practical examples and should
use appropriate vocabulary.
Children can identify
outcomes and can justify their
predicted outcomes in simple
terms.
Questions such as these will
depend upon the location and
the time of year.
In a given situation, children can
compare objects/events to
determine which is longer, taller,
heavier, hotter, larger and can
demonstrate direct comparisons:
“My pencil is longer than his”,
“Her bag is heavier than mine.”
Young children need many
opportunities to experience
and quantify measurement in
a direct kinaesthetic manner.
Later understanding of
measurement will be based on
this foundation.
Children are given
opportunities to directly
compare three or more objects
of unequal length.
Children can: put three trains
in order with the longest first;
put three bears in order of size
starting with the shortest; with
three friends, find who has the
longest feet, hair or socks.
Children should also be given
objects of equal length to lead
to the idea of equivalence.
Are objects still the same
length even when they are not
lined up next to each other?
Children play with sand and
water using a variety of
everyday containers.
Children can describe which
containers are full/empty and
which containers hold more
than others.
Experiences of filling
containers with sand and
water will eventually lead
children to an understanding
of capacity/volume. However,
this is not a specific
expectation at this time.
Children are supplied with
three or four containers of the
same size filled with different
objects: cotton wool, beads,
modelling clay. Children are
asked to arrange the containers
in order from lightest to
heaviest (or vice versa).
Children can put the
containers in the correct order
and can describe how they
found this order. They
recognize that mass is not
directly related to the size of
the container.
All assessments done at this
stage will be oral and
practical in nature.
Observations may be noted by
the teacher.
Children can sequence events
correctly and can explain that
it will be time to go home
after a story, time for bed
after bathtime etc.
Discussion of familiar timerelated events provides the
first stage in understanding
time.
Children will:
• discuss and identify
outcomes that will happen,
won’t happen and might
happen.
What will happen if…?
Is that the only thing that
might happen? perspective
Probability questions
Children are asked a series of
questions: Will an elephant
walk in through the classroom
door? Will a person walk in
through the classroom door?
Will it snow in mid-summer?
Will it be sunny in midsummer? Will it be dark at
lunchtime? Will it be light at
lunchtime? Will it be your
birthday tomorrow? Will it be
Thursday tomorrow?
In discussions about the
weather, children are asked a
series of questions related to
probability: Will it rain
today? Will it snow today?
Do you think the sun will
shine all day? Will this storm
stop?
Measurement
To measure is to attach a
number to a quantity using a
chosen unit. However, since
the attributes being measured
are continuous, ways must be
found to deal with quantities
that fall between the numbers.
It is important to know how
accurate a measure needs to
be or can ever be.
• identify, compare and
describe attributes of real
objects and situations:
longer, shorter, heavier,
empty, full, hotter, colder
Which one is longer, taller,
heavier, hotter?
Can you show me which is
longer, taller?
Which holds more?
How can you show me?
How do we know it is
heavier?
• identify, compare and
sequence events in their
daily routine: before, after,
bedtime, storytime, today,
tomorrow.
How will we know when it is
time to go home? function
What will happen before…?
What happens after…?
connection
Comparisons
Children are given real
objects and situations in
which they must compare
length, mass, duration and
temperature.
Yesterday, today and
tomorrow
Children draw pictures of
what happened yesterday,
what is happening today and
what they hope will happen
tomorrow. They explain the
sequence of events to another
child.
Direct and indirect
comparisons lay a foundation
for conservation.
Many children’s storybooks
contain references to time and
to sequences of time.
Why do we put our socks on
before our shoes?
What is your favourite part of
the day and why? reflection
3.18
Content
Shape and space
Children need to understand
the interrelationships of
shapes, and the effects of
changes to shape, in order to
understand, appreciate,
interpret, and modify our twodimensional and threedimensional world.
Page 3 of 6
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Children can sort and describe
the shapes and talk about how
they worked.
Young children need many
opportunities to experience
and quantify space in a direct
kinaesthetic manner.
Children will:
• sort, describe and compare
3-D shapes according to
attributes such as size or
form
What is the same about these
shapes?
What is different about them?
function
What is the same about
shapes that are flat/curved?
How can we use these
shapes? responsibility
What can you tell me about
your shape? reflection
Why is a block the best shape
for building a tower?
causation
Exploring 3-D shapes
Children are given a variety
of 3-D objects that may
include geometric shapes.
They are encouraged to look
at the objects in a variety of
ways, and from different
directions, to identify
similarities and differences.
Children can name some of
the shapes and begin to
classify them according to the
shapes’ attributes.
Children are given a variety
of shapes and objects to
explore by touching,
observing and talking about
them.
Young children often surprise
us with the creative categories
they devise for sorting. It is
important to realize that, at
their development level, this
is completely appropriate.
Children explore whether
objects/shapes roll or stack.
3-D shapes are the real-life
objects with which children
are familiar, whereas 2-D
shapes (plane shapes) are
actually an abstraction. For
this reason it is recommended
that 2-D shapes be introduced
as faces of 3-D shapes
through drawing around,
painting and printing with 3D shapes.
Can we make this shape from
other shapes? causation
Does a shape change if its
size changes?
Does a size change if its
shape changes?
Does it always look the same
when you turn it upside
down/over?
The regions, paths and
boundaries of natural space
can be described by shape.
To identify pattern is to begin
to understand how
mathematics applies to the
world in which we live. The
repetitive features of patterns
can be identified and
described as generalized rules
called functions. This builds a
foundation for the later study
of algebra.
• explore and describe the
paths, regions and
boundaries of their
immediate environment
(inside, outside, above,
below) and their position
(next to, behind, in front of,
up, down).
• find and describe simple
patterns
How can we describe
position?
What is inside?
What is outside?
What is the same and what is
different about inside and
outside?
Can we see any patterns in
this picture?
Can we describe these
patterns?
Regions and boundaries
Children use ropes, walls etc
to explore regions and
boundaries and their position
in relation to these objects.
In PE, children stand behind,
in front of or next to a partner.
Children use rubber bands
and geoboards to create
shapes with defined
boundaries.
Finding patterns
Children are given pictures to
look at that may/may not
include simple patterns.
They discuss and describe any
patterns they see.
Can we describe the pattern
in another way?
What is a pattern? form
Children need to build, move
and play with shapes in order
to understand the shapes’
attributes. Later
understanding of shape will
be based on this foundation.
Children can use everyday
language to describe what is
inside, outside, next to, in
front of and behind in a given
situation.
Children can use their own
language to describe their
position in relation to others.
Children can use their own
explanations to describe the
patterns they see and can
identify and describe the
repetitive features.
Children can predict the next
step in a pattern.
Children recognize patterns in
animals: stripes, spots,
feathers, wings of butterflies.
Children recognize patterns in
carpets, architecture and
common household objects.
Children can identify the
common features of each
pattern and are able to
compare the patterns to
identify similarities and
differences.
The world is filled with
pattern. Children should find
and describe patterns in art,
physical education, music,
literature etc. Encouraging
children to identify patterns
across the curriculum embeds
this important idea.
Children will describe the
in their own way.
Children sort, match and
compare patterns on scraps of
wallpaper or fabric.
3.19
Content
Pattern and function (cont.)
Page 4 of 6
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Children will:
• create simple patterns using
real objects.
What pattern can we make
with these shells, buttons,
blocks? function
Creating patterns
Children use blocks, colour
tiles, buttons or keys to copy
and create simple patterns.
Children can create their own
patterns or identify if two
patterns are the same.
Can we make a different
pattern with them?
Can we make up a rhythm
pattern using our hands and
feet? function
Number
Our number system is a
language for describing
quantities and relationships
between quantities. The value
attributed to a digit depends
on its place within a base
system. The operations of
addition, subtraction,
multiplication and division are
related to one another and are
used to process information in
order to solve problems.
• read, write and model
numbers to 20
What numbers do we know?
Where do we find numbers?
Create a rhythm pattern using
clapping, clicking fingers,
stamping feet. Children begin
to hear simple rhythms in
music.
Number walk
Take the class on a number
walk around the classroom,
school buildings or school
grounds. Children start to
point out numbers in the
immediate environment and
use them to create a display of
“numbers we can see.”
Reading and writing
numbers
Children use numeral cards to
label sets of objects.
Children draw numbers in
sand with their fingers, paint
numbers and make numbers
using modelling clay.
• count, compare and order
numbers to 20
How can we find out how
many things are here?
Counting activities
Children count objects
arranged in a line or
randomly.
Links to musical patterns,
using percussion instruments,
can be highlighted in this
context.
Children can recognize
numbers in their immediate
environment and can read the
numbers on a clock face or on
a computer keyboard.
Children can write the number
that is represented when they
have counted objects.
Children can reliably count a
given set of objects.
What number comes next?
What was the number before
that?
Children count the number of
jumps it takes to cross the
room.
A wide variety of everyday
materials should be used, as
well as some mathematics
manipulatives such as Unifix
cubes.
Children count and compare
the letters of their names.
Rockets taking off require a
countdown. Children start
from numbers other than ten
and use zero as part of the
sequence.
Assessment will be oral in
nature.
Children should have richly
varied opportunities for
counting real objects that are
related to their studies.
Children count drum beats or
claps in a rhythm.
Number rhymes and songs
Children join in with rhymes
and songs such as Five Little
Speckled Frogs, Ten Green
Bottles and One Man Went
To Mow.
The teacher could make
number labels for seats, coat
pegs etc for children to find
on their number walk.
Children can join in with
rhymes and songs that involve
counting to and from 10.
There are many finger and
action rhymes suitable for
developing an awareness of
numbers and counting.
Children can count forwards
and backwards to and from
10.
3.20
Content
Number (cont.)
Page 5 of 6
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Children will:
• estimate quantities to 10
How is estimating the same as
counting?
How is estimating different
from counting? connection
How many objects do you
think I have in my (closed)
hand?
How many objects do you
think I have in my (open)
hand now?
Did you change your
estimate? Why?
Which is easier to estimate?
Why?
• use ordinal numbers to
describe the position of
things in a sequence
order in which these things,
objects, children are
arranged?
Estimating to 10
Pick up a few objects in a
closed fist and ask the
children to guess how many
objects are there. Open the
hand and ask them to guess
again. Ask them to count the
objects to check their estimates.
What connections are there
between these numbers?
connection
How does the number change
when we add one more?
change
• use the language of
mathematics: more, less,
number names, total
What number names do we
know?
What is a total?
Compare the ease of
estimating when children
can/cannot see the objects.
They can check their estimate,
and refine it if necessary, after
counting.
Estimation is a skill that will
develop with experience and
will help children gain a
“feel” for numbers. Children
must be given the opportunity
to check their estimates so
that they are able to further
refine and improve their
estimation skills.
Place counters, sweets, coins,
marbles or other small objects
on a dish. Ask a group of
children to estimate how
many objects are there. Give
them the opportunity to
change their estimates if
necessary. Count the objects
and then discuss who was the
nearest. Repeat with different
objects and with different
children counting. Gradually
increase the number to 20.
Places in the races
Children draw a picture based
on the following information:
In the Farmyard Olympics
Sprint, horse came first, pig
was second, and goat was in
fourth position behind rooster.
Where in the line?
Use cardinal numbers on cards
to label a line. These can later
be replaced with ordinal
numbers. When children arrive
in the classroom, they peg up a
photograph of themselves in
the order in which they arrived.
Start with labels up to 5, then
10, and up to 20 after practice.
• model number relationships
to 10: “Show me one more
than three, take two away
from these cubes”
Children can make reasonable
estimates of small groups of
objects without counting.
Magic numbers
Use a stick of 10 linked cubes.
Out of sight of the children,
remove a certain number of
cubes and keep them hidden.
Show the remaining cubes to
the children and count them.
Ask if they can find the
missing magic number.
Frequent sessions of this kind
quickly build up number–bond
recall.
Mathematical vocabulary
Model the language of
mathematics with which
children may not be so
familiar: more than, less than,
fewer, same as.
Encourage children to talk
about the numbers, and the
mathematical words they
know, in their painting,
drawing, food preparation and
role play.
Try to find authentic contexts
in your units of inquiry to
practise estimating up to 20.
Children can create pictures
that accurately represent the
information in the story.
Arrange athletic activities
(jumping, throwing, running
events) to allow children to
use the words “first”,
“second”, “third” in context.
Children can describe orally
the order in which they
arrived in the room, using the
correct ordinal numbers.
For a child who knows
cardinal numbers, learning
ordinal numbers is primarily a
language issue.
Children will also have to
demonstrate an understanding
of positional language:
before, after, between, next.
Children can show the
different ways in which they
can make addition sentences
using manipulatives that equal
4, 8, 10.
Good number sense is based
on richly interrelated number
meaning.
Children can use these new
terms in their discussions.
From the earliest stages,
correct and appropriate
mathematical language should
be used. Consistent use will
enable children to remember,
and use, relevant mathematical
vocabulary with confidence.
Children can use number
words and mathematical
vocabulary in their play:
“There are four plates”, “I am
four!”, “There are more dogs
in my painting than cats”,
“I have the same number of
pencils as you.”
Mental arithmetic, including
the basic facts, is easier if
children have had experience
of deconstructing numbers:
8 = 5 + 3 = 6 + 2.
3.21
Content
Number (cont.)
Page 6 of 6
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Children will:
How will we count these?
• use 1–1 correspondence
How can we be sure that we
have the right number of
plates/cups?
• explore the conservation of
number through the use of
manipulatives
(■ ■ ■ ■
= ■
■
■
■)
How many objects are here?
How many are there now?
Is it the same number?
How do you know?
• select and explain an
appropriate method for
solving a problem.
How can we find an answer?
Are there any other ways to
find out?
What (materials) can we use
to help us?
Do we always find the same
answer even when we do
things in a different way?
change
Which way works the best?
Why? reflection
One to one matching
Children match objects using
1–1 correspondence.
Children set the table for four
friends in the home corner.
Conservation of number
Act out the story of The Three
Billy Goats Gruff. Children
can see that even when one of
the goats “trip-traps” over the
bridge, there are still three
goats.
Children can lay the table to
correspond to one plate or cup
or set of cutlery for each
person.
Children can identify the fact
that there are still three goats
regardless of their position.
Ask six children (or any other
number) to stand in a group.
They may be holding numeral
cards. When they are bunched
together, count and identify
all six members of the group.
Ask the children to spread out
from each other and count
again. They can regroup in
any way to demonstrate the
conservation of number. This
activity can be repeated with a
variety of different objects.
Doing it my way
Give the children a problem,
and ask them to select
materials to help them solve
it. Children explain why they
chose those materials.
To conserve, in mathematical
terms, means the amount stays
the same regardless of the
arrangement. Young children
are often fooled by what their
eye sees when objects are
rearranged.
Lots of counting and
rearranging of manipulatives
will help in overcoming this.
Children cannot be taught
how to conserve, but an
understanding is reached
through reason and
experience.
Conservation can also apply
to measurement relationships.
Children can suggest a way of
finding an answer to a
problem.
the method selected may not
be the most effective.
Children can explain why
their method will work.
Children should have regular
opportunities to select and
explain their ideas and
methods.
Children can select
manipulatives to help solve a
problem.
Children who have been
encouraged to select their
own apparatus and methods,
and who become accustomed
to discussing and questioning
their work, will have
confidence in looking for
alternative approaches when
an initial attempt is
unsuccessful.
Children notice and discuss
difficulties and make
suggestions to one another.
3.22
Page 1 of 10
Students will have the opportunity to identify and reflect upon “big ideas” by making connections between the questions asked and the concepts that drive the inquiry. They will become aware
Students will sort, label, collect, display and compare data in a variety of forms, including pictographs and bar graphs. They will understand the purpose of graphing data. They will discuss,
identify, predict and place outcomes in order of likelihood.
Measurement
Students will estimate, measure, label and compare using non-standard units of measurement, and understand why we use standard units of measurement to measure length, mass, time and
temperature. They will read and write time to the hour, half hour and quarter hour, and identify and compare lengths of time (days, weeks and months).
Shape and space
Students will describe the properties of 3-D shapes, including the 2-D shapes that can be seen, using mathematical vocabulary. They will find and explain symmetry in the immediate
environment and create symmetrical patterns. They will give and follow simple directions using left, right, forward and backward.
Students will describe, continue, create and compare patterns. They will recognize and extend patterns in number. They will identify commutative property. They will model the relationships
in, and between, addition and subtraction.
Number
Students will read, write, estimate, count, compare and order numbers to 1000. They will read, write, model and understand addition and subtraction, using mathematical vocabulary and
symbols. They will automatically use addition and subtraction facts to 10. They will explore multiplication and division using their own methods, use fraction names to describe part and whole
relationships, and explore counting patterns. They will select and explain appropriate methods for solving a problem and estimate reasonableness of answers.
Content
How best will students
learn?
students have learned?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and transdisciplinary
skills
expectations. Students should be
Students will:
• sort and label objects into
sets by one or more
attribute
Can we show how these
objects are the same and how
they are different?
How do we decide which
objects to put into which sets?
Can we explain how we
sorted our objects?
How can one object be part of
two different sets?
Sorting by attributes
Students sort pattern blocks
into sets and label the sets.
Adding other objects to the
collection of pattern blocks
and asking the students to sort
and label the new sets.
Students can determine the
attributes by which sets are
sorted: “These objects are all
round and blue.”
As each pattern block has a
specific size, shape and
colour, students can sort them
by more than one attribute.
Students look at the various
attributes of each of the
pattern blocks.
Students will require a lot of
practice of sorting and
labelling objects into sets by
more than one attribute:
“Everything in this set is
triangular and green.”
Can we show other ways of
sorting these objects?
• discuss and compare data
represented in teachergenerated diagrams: tree,
Carroll, Venn
Why would we want to use
diagrams to represent our
data?
What information can we tell
by looking at these diagrams?
How can we tell that data
belongs in more than one set?
How do we know if the same
information is represented in
both of these diagrams?
students have had a lot of
hands-on experience of
sorting and labelling objects
into sets by one common
attribute.
Comparing data
Students are presented with
data previously sorted in a
Venn diagram. They are asked
to determine how the data in
each set is sorted. Students are
given several objects and are
asked to add them to the
appropriate set and to justify
their reasoning.
Students are given a Venn
diagram and a Carroll
diagram representing the
same information. They can
explain whether or not the
diagrams represent the same
data and how they know this.
It is easier for students to
understand and interpret sets
and subsets that intersect if
somebody else has created
them. Being able to create
diagrams, subsets and
intersecting sets is more
complex and challenging.
Computer programs that
require users to manipulate
data in order to create graphic
organizers are excellent tools
for developing an
understanding of these
mathematical concepts.
3.23
Content
(cont.)
Page 2 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students will:
• collect, display and
interpret data for the
purpose of finding
information
Why would we want to collect
data?
What questions will give us
the information we need?
How can we display data?
What information can we tell
by looking at these diagrams?
How can we tell that data
belongs in more than one set?
Collecting and displaying
data
Students work in groups to
find the answer to a question:
How many students have
pets? What pets do they have?
collect and record the data in
any form.
Students display their data in
two different forms: list, tally
marks. They exchange their
data displays with another
group and interpret the
findings.
Students are given a set of
data. They can interpret the
information and explain how
the information can be used.
Students can represent the
same data in a different
format.
Interpretations of data should
include the information that
cannot be concluded as well
as that which can. It is
important to remember that
the chosen format should
without bias.
How do we know if the same
information is represented in
both of these diagrams?
• understand the purpose of
graphing data
Why would we want to make
a graph?
How are graphs used?
How are graphs the same as,
and different from, lists and
other types of diagram?
What are the important
features of a graph?
How did we reach our
interpretations? reflection
• create a pictograph and
simple bar graph from a
graph of real objects, and
interpret data by comparing
quantities: more, fewer, less
than, greater than
How can we keep a record of
our real-life graph?
What are the similarities and
differences between real-life
graphs, pictographs and bar
graphs? function
How can we interpret the data
from these graphs and
describe our findings to other
people?
Do all of these graphs give us
the same information?
What information can/cannot
be determined by looking at
the graph?
Why do we use graphs?
samples of real-life graphs,
simple pictographs and bar
graphs. They are asked to
look for similarities and
differences between the
features and set-up of each
graph.
Students interpret the data
from each graph, including
the information that can be
determined as well as that
which cannot.
Creating simple graphs
Students sort and label their
snacks into different
categories and use the empty
wrappers to create a real-life
graph based on their findings.
Students use the real-life
graph to create a bar graph
representing the same data.
They provide a title for their
graph and label each axis.
Students interpret the data and
compare quantities.
Students need to think about
the data they have collected
and how they can organize it
in the most effective and
efficient way. They should
have a lot of experience
organizing data in a variety of
ways, and of talking about the
advantages and disadvantages
of each.
Students can record data as a
pictograph and in other
formats: list, tally marks.
Graphs are used to represent
relationships and to highlight
Students can explain which
display they think best
represents their data and why.
Students need opportunities to
collect, organize, display and
interpret data.
Students can give examples of
how graphing data helped
them to answer questions.
Students can create a
pictograph from a graph of
real objects.
Students are given a
pictograph and can create a
bar graph representing the
same data.
Different types of graph
highlight different aspects of
data more efficiently. In order
to interpret data effectively,
students need practice in
determining information that
can/cannot be gained by
looking at particular graphs.
Based on their interpretations
of the data, students can write
three statements that are true
and three statements that are
false.
students can begin to use
squares or tiles to record data.
These can be simply viewed
as the frame of a pictograph.
Students can compare and
contrast the two displays,
including information that
cannot be determined.
3.24
Content
There are ways to find out if
some outcomes are more
likely than others. Probability
can be expressed qualitatively
by using terms such as
certain, likely, unlikely or
impossible.
Age range: 5–7 Years
Page 3 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students will:
• discuss, identify, predict
and place outcomes in
order of likelihood:
impossible, unlikely, likely
and certain.
How do we know whether or
not something is likely to
happen?
Why are some things more
likely to happen than others?
How can we decide?
How can we describe and
compare the likelihood of
something happening?
Can it happen?
Students discuss situations,
from stories read in class, in
terms of the likelihood of their
happening in real life:
children can fly, dreaming
about flying.
Students are given a list, or
pictures, of situations and can
place them in order of their
likelihood, from impossible to
certain. They can explain and
defend their thinking.
Students label these situations
as impossible to happen,
unlikely to happen, likely to
happen or certain to happen in
real life. Students defend their
answers.
• estimate, measure, label
and compare using nonstandard units of
measurement: length, mass,
time and temperature
When would we want to
measure something? function
How can we find out how
long/heavy something is?
How can we know if one
thing is longer/heavier than
other things we have
measured?
Why would we want to
estimate the measurement of
something?
Measuring using
non-standard units
Students measure, record and
compare the length of
different objects in the
classroom using non-standard
units of measurement. They
label their measurements with
the unit used: “The book is
eight paper clips long.” Based
on their findings, they
estimate the length of other
objects in the room and justify
their estimations. Students
measure the objects and
compare their estimations to
the actual measurements.
How can knowing about
measuring help us to make
good estimations?
In how many ways can these
objects be measured?
Students can reasonably
estimate the length, mass
and/or temperature of given
items using non-standard
units. They can explain how
they determined their estimates
and how they can check them.
Students can check their
estimations by actually
measuring the objects.
Based on their findings, they
can estimate the length, mass
and/or temperature of other
items and explain how they
determined their estimates.
Students are given 3-D
objects: oranges, apples,
stones. They measure and
compare them.
Students can measure
circumference with string and
measure the weight, in nonstandard units, using a
balance scale.
Students are given a selection
of mystery parcels to be
posted. They put them into
order by mass. The parcels
can be put into plastic bags
and held by the fingertips to
avoid the size of the parcel
influencing students’
judgments.
order the parcels. With
practice, their accuracy and
speed will increase.
Sand and water activities that
involve the use of nonstandard units of
measurement are provided:
How many egg cups full of
water fill the jar? How many
spoonfuls of sand fill the
yogurt pot?
Discussions need to take place
in which students can share
their sense of likelihood in
terms that are useful to them.
These discussions need to
occur repeatedly.
Our sense of probability often
defies mathematical definitions
and should not be dismissed. A
good example is flipping a coin.
Although every time you flip a
coin the mathematical chance
that it will fall on one side rather
than the other is equal, many
people would say that, after a run
of one side, the chance of
flipping the side we have not yet
seen increases. There is some
common sense in this.
Comparing events that happen
in the real world provides
opportunities for students to
clarify their ideas.
Measurement
are continuous, ways to deal
with quantities that fall
between the numbers must be
found. Also, by attaching a
number to a quantity, students
can begin to understand how
real world.
Situations that come up
naturally in the classroom,
often through literature,
present opportunities for
discussing probability.
The process of measuring is
identical for any attribute:
choose the unit, compare that
unit to the object and report
the number of units.
Students develop an
understanding of
measurement by using
materials from their
immediate environment as
measuring units such as corks,
beads and beans. They begin
to investigate how units are
used for measurement and how
measurements vary depending
on the unit that is used.
Students will refine their
estimation and measurement
skills by basing estimations on
prior knowledge, measuring
the object and comparing the
actual measurements to their
estimations. Developing
personal referents will help in
this process.
Experiences of filling
containers with sand and
water will eventually lead
students to an understanding
of capacity/volume. This,
however, is not a specific
expectation at this time.
3.25
Content
Measurement (cont.)
Page 4 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students will:
• understand why we use
standard units of
measurement to measure
What tools can we use to
measure?
Does the measurement of the
same object change when we
use different measuring tools?
change
How can we compare and
describe the measurement of
two objects so that other
people can understand?
• use a calendar to determine
the date, and to identify and
sequence days of the week
and months of the year
Why do we use calendars?
How can we use a calendar to
find out the date?
What patterns are there on the
calendar? form
Why is it important to be able
to put the days of the week
and the months of the year in
the correct order?
responsibility
• estimate, identify and
compare lengths of time:
second, minute, hour, day,
week, month
What language do we use to
describe time? form
Why would we want to
measure time?
How do we measure time?
function
What can you do in a
second/a minute/an hour?
How do we know when a day/
a week/a month begins and
ends?
Why do we use standard
units?
Students measure, label and
compare the results of
measuring the same object using
different non-standard units
such as their hands and pencils.
Students explain why the
measurement may change even
though the object remains the
same.
Using a calendar
Students use the classroom
calendar as a reference for
setting up their own monthly
calendars at the beginning of
each month. They fill in the
name of the month, the year
and the days of the week in
the correct order.
Students fill in the date on a
daily basis and discuss different
aspects of the calendar: Which
day of the week comes
between Monday and
Wednesday? How many
months away is the month of
August?
How long will it take?
Students estimate how many
times they can write their
name in one minute. Set a
timer. Students repeat the
activity estimating how many
times they can jump up and
down in a minute. They
justify why the number of
jumps is more or less than
their estimate of the number
of times they wrote their
name.
How can counting by 5 and
10 help us to tell the time?
fractions help us to tell the time
Why is time referred to in
quarter hours and half hours?
Telling the time
Students use digital and
analogue clocks to tell time to
the quarter hour, half hour and
hour.
Informal and personal systems
of measurement need to
become standardized so we can
compare measurements
accurately and communicate
measurement in a consistent
way.
be or can ever be.
Students make their own tools
for measuring.
Students predict how long they
think it will be until lunchtime,
based on how many half hour
intervals will occur during this
time.
Set and reset a timer for 30
minute intervals, keeping a
record of every time the timer
goes off until lunchtime.
• read and write the time to
the hour, half hour and
quarter hour.
Students can explain why two
students may get different
answers when they measure
the same space in the
classroom with their feet.
Using the classroom calendar,
students can indicate the
current date, day of the week,
month of the year, date of the
month, year and any special
days: birthdays, holidays.
Students will need a lot of
real-life practical experiences
to develop an understanding
of time.
Time is an abstract, intangible
form of measurement.
Students can state the date
before and after the current
date: today is, yesterday was,
tomorrow will be. Students
can use their completed
calendars to answer questions:
What was the date of our field
trip to the farm?
Students can compare their
estimates to the actual number
of times they wrote their
name.
Students need to understand
time in terms of the duration of
time (an hour) and the specific
time (3:12pm).
These are two very different
notions. Daily calendar
activities and classroom
schedules will help develop
these concepts.
Repeat these activities several
times over a few days until
students have an
understanding of how long a
minute is.
Students can compare their
estimates with the actual time
that elapsed. Students predict the
time left until the end of the
school day. They can justify why
their estimate for the time
remaining in the school day is
more or less than previous
estimates.
Students are given a specific
time and can write what the
time will be 15 minutes from
then, an hour from then and a
half hour from then.
Being able to tell the time is a
concept that is often pushed
too early. Students need to
understand the duration of
time before being introduced
to the moment of time.
How do we record the time of
day?
3.26
Content
Shape and space
interpret and modify our twodimensional and threedimensional world.
Page 5 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students will:
• use what they know about
3-D shapes to see and
describe 2-D shapes
How can knowing about 2-D
shapes help you to learn more
about 3-D shapes?
How are these shapes the
same as, and different from,
each other?
Which 2-D shapes make up
this 3-D shape?
Can you explain how the
cereal box was made? change
• sort and label 2-D and 3-D
shapes using appropriate
mathematical vocabulary:
sides, corners, circle,
sphere, square, cube
What are these shapes like?
form
How are they the same as, and
different from, each other?
function
What are the properties of
similar 3-D shapes?
What mathematical names do
we give to these shapes?
How many different ways can
we sort these shapes?
Finding 2-D faces on 3-D
shapes
Students trace around, paint
and print from the faces of
3-D shapes.
Students deconstruct 3-D
shapes into 2-D faces.
Students cut up a 3-D
container in order to see the
2-D elements. They sort and
mark the 2-D pieces
according to shape. For
example, colour all of the
squares red and all of the
rectangles blue. Then they
reconstruct the container and
describe the 2-D aspects of it.
Sorting 2-D and 3-D shapes
Students sort 2-D and 3-D
shapes (real objects and
geometric shapes) into
different sets. They discuss
what is the same/different
about the shapes and describe
the properties.
Students explain why they
sorted the shapes in the way
that they did and provide
labels for each set based on
their explanation.
Looking at a student’s work,
other students describe the
sets and provide labels for
each set.
Students look for examples of
shapes in the immediate
environment and explain how
they recognize them by their
characteristics.
What words do
mathematicians use to describe
parts of shapes?
• create 2-D shapes
What do you know about 2-D
shapes?
How do you know if a shape
is a 2-D shape?
Where can you find 2-D
shapes?
Provide opportunities for
describing real objects using
mathematical vocabulary:
side, corner, diagonal.
Creating 2-D shapes
Students are given a
description of a 2-D shape
and are asked to create the
shape using different
materials: paper, pencil, tape,
paint, geoboards.
When given a 3-D shape,
students can describe the 2-D
elements of it. They can create
2-D shapes and assemble
them to create a 3-D shape.
2-D shapes, in reality, only
exist as faces of 3-D objects.
We help students understand
the concept of planes by
teaching 3-D before 2-D.
Students can draw a 2-D and
3-D representation of a 3-D
shape.
Computer programs that
create 2-D and 3-D shapes
would be excellent tools to
use for this unit of study. This
takes away the pressure of
students having to draw the
objects themselves while
building on spatial awareness
and 2-D and 3-D properties.
Students can describe how
their 2-D drawing helped
them construct their 3-D
drawing.
Given a description of a 2-D
or 3-D shape, students can
create it and name it.
Students can describe the
properties of a pre-organized
set of shapes.
attributes by which a set was
sorted and can label the set
accordingly. They can find
other shapes to add to the
collection.
Students can sort shapes into
many different sets. Each time
they can explain how they
sorted each set and can
provide an appropriate label.
There must be a wide variety
of materials available for
students to construct 3-D
shapes from 2-D faces.
Through creating and
manipulating shapes, students
align their natural vocabulary
with more formal
mathematical vocabulary and
they begin to appreciate the
need for this precision.
the properties of 2-D and 3-D
shapes before the
mathematical vocabulary
associated with shapes makes
sense to them.
Students can point out a
variety of 2-D and 3-D shapes
in the immediate environment
and describe the attributes of
each shape.
Students can use the correct
mathematical vocabulary to
describe shapes.
Students can create a 2-D
design by creating and
combining as many 2-D
shapes as possible.
Drawing 2-D shapes is a
much easier task than drawing
3-D shapes. 3-D shapes
involve visual perception and
spatial awareness skills that
may be very challenging for
students of this age.
What materials can we use to
create 2-D shapes?
3.27
Content
Shape and space (cont.)
Page 6 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students can find examples of
symmetry in their immediate
environment. They can
explain what makes
something symmetrical and
what would cause them to be
asymmetrical.
Symmetry is an important way
of understanding and
comparing shapes.
Students can use pattern
blocks to create a symmetrical
design and can check the lines
of symmetry by using a
mirror. They can explain why
their design is symmetrical.
Students need to participate in
numerous activities, involving
folding paper and using
mirrors, to explore and
understand lines of symmetry.
Students will:
• find and explain symmetry
in their immediate
environment
Where can you find examples
of symmetry?
How do you know if
something is symmetrical?
How can you find a line of
symmetry?
Finding symmetry
symmetry in magazines and
architecture. Advertising
logos and car wheels are often
good examples of
symmetrical designs. Students
collect and discuss the
examples they find.
Students will need to have a
lot of hands-on experience in
order to develop an
understanding of symmetry.
Does anyone have a different
way of explaining this
pattern? perspective
• create and explain simple
symmetrical designs
Is this pattern symmetrical?
What materials can you use to
create a symmetrical design?
How do you know whether or
not your design is
symmetrical?
Can we make an asymmetrical
design symmetrical and vice
versa?
• give and follow simple
directions, describing paths,
regions and boundaries of
their immediate environment
and their position: left, right,
forward and backward.
When would we need to
give/follow directions to get
somewhere?
How can you describe your
position to somebody who
cannot see you?
How precise do directions
need to be?
Creating symmetrical
designs
With a partner, students use
cubes and grid paper to create
a symmetrical design. They
fold the grid paper in half to
create a line of symmetry. One
student places a cube in a
square on the grid, the other
student has to place a cube in a
square on the other side of the
line of symmetry to make the
design symmetrical. Students
use mirrors to verify that their
design is symmetrical.
Following directions
Students create a maze in the
playground or by moving the
furniture around in the
classroom. They are paired up
with another student and take
turns blindfolding each other.
They give specific directions
to their blindfolded partner in
order to move them through
the maze.
Students can colour in the
squares on the grid paper to
keep a record of their designs.
Students can draw a map of
their classroom and can
describe where certain things
are in relation to a central
point: left, right, forward,
backward.
Many people confuse their
right from their left. Students
need many opportunities to
experience and quantify space
in a direct kinaesthetic
manner.
Students can find, describe
and extend a pattern in their
immediate environment.
The world is filled with
patterns. Having students
identify patterns across the
curriculum embeds this
important idea. Students will
describe the repetitive features
of patterns in their own way.
to understand how
of algebra.
• create, describe and extend
patterns
Where do we find patterns?
What do you know about
patterns?
How can patterns help us?
In what ways are these
patterns similar and/or
different?
What can go in the missing
space?
Is that the only possible
cube/colour?
In how many different ways
can you describe this pattern?
Creating and extending
patterns
Students are given a pattern
created with pattern blocks.
They are asked to describe the
pattern and to extend it.
Using coloured cubes, or
squares, the teacher makes a
pattern and leaves spaces for
the student to complete.
Students prepare patterns for
each other to complete. Later,
substitute a cloud shape or
box in the space to be filled.
Using a variety of
manipulatives, students create
and describe two different
patterns. They compare
whether their two patterns are
the same and justify their
answer.
They can justify how they
know it is a pattern.
Students are given a label
such as AABCC and can
create and describe a pattern
following this form.
They can create and describe
a second pattern, following
the same form, and can show
that the two patterns are the
same.
Using pattern blocks for
patterning activities helps
students to see that one object
can have more than one
attribute within a series of
patterns. Each pattern block
could be categorized
according to shape, size and
colour.
3.28
Content
Page 7 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students can describe, explain
and extend any pattern they
find in the 100’s chart.
Students turn their patterning
skills to the numbers they
already know. The patterns
they find will help to deepen
their number concepts and to
extend their range of numbers.
Students will:
• recognize, describe and
extend patterns in numbers:
odd and even, skip
counting, 2s, 5s and 10s
What patterns do you notice
in the numbers you know?
How can we know what
number comes next in a
sequence?
How can working with
number patterns help us?
Patterns in numbers
Students model a number
pattern using manipulatives.
(They have a group of 2
cubes, then a group of 4, then
6, to represent counting by 2.)
They are asked to describe,
explain and extend another
student’s number pattern.
This can be done orally, with
the use of manipulatives,
and/or in written form through
pictures, equations or
colouring in the pattern on a
100s chart.
Using a number line, students
mark the various patterns of
skip counting. The 2s can be
underlined, the 3s put into
triangles and the 4s put into
squares.
• identify patterns and rules
for addition: 4 + 3 = 7,
3 + 4 = 7 (commutative
property)
How are 4 + 3 and 3 + 4
connected? connection
In what ways are 4 + 3 and
3 + 4 the same?
All numbers make patterns
and it is the use of patterns in
number that makes
mathematics so powerful.
Algebra takes the patterns in
number and uses them to
explain many things that
would otherwise remain a
mystery.
Given a number line, students
know if 30 is in the skip
counting sequence of the 5s,
and what the next five numbers
are in this sequence. They can
explain how they know this.
Patterns and rules for
addition
Students model 5 + 2 and
2 + 5. They use their models
to compare and contrast the
two equations.
Students can make a model to
show how 4 + 3 and 3 + 4 are
connected. They can explain
the patterns and rules used.
Many students intuitively
work out the commutative
property. It is important that
they understand that, while it
doesn’t matter in which order
sets are added as the answer
remains the same, 4 + 3 is
generated by a different story
to 3 + 4.
subtraction
Students model 7 – 3 and
7 – 4. They use their models
to compare and contrast the
two equations.
show how 7 – 3 and 7 – 4 are
It is important for students to
explore the patterns and rules
for addition and subtraction
through problem solving.
In what ways are 4 + 3 and
3 + 4 different?
Why does the sum remain the
same even if the addends
appear in a different order?
causation
Why do these calculations
produce patterns? causation
for subtraction: 7 – 3 = 4,
7–4=3
How are 7 – 3 and 7 – 4
connected?
In what ways are 7 – 3 and
7 – 4 the same?
Real-life contexts should be
used as often as possible.
In what ways are 7 – 3 and
7 – 4 different?
Why is the order of numbers
in a subtraction equation so
important?
What effect does the order of
numbers have on the difference?
• model, with manipulatives,
the relationship between
addition and subtraction:
3 + 4 = 7, 7 – 3 = 4.
How are addition and
subtraction connected?
connection
addition that can help you
with subtraction?
How can knowing your
addition facts help you with
your subtraction facts?
subtraction related?
Students are given the
numbers six, two and eight
and are asked to model as
many addition and
subtraction equations as they
can, using all three numbers.
Students model and write the
equations. They determine
and explain the connection
between addition and
subtraction.
show how 3 + 4 and 7 – 3 are
Besides introducing the
important mathematical
concept of inverse operations
in a user-friendly context,
finding “families” of number
facts is another way for
students to learn and
remember the subtraction
facts.
3.29
Content
Number
on its place within the base 10
order to solve problems. The
degree of precision needed in
calculating depends upon how
the result will be used.
Page 8 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students can explain how the
base 10 system works to
another student.
In English, the numerals 10 to
19 are harder to encode and
decode than 20 to 99 since the
language is less regular. 24 is
written in the same order as it
is said whereas with 17, the
second digit is heard first.
There is also confusion
between the teens and the
tens.
Students will:
numbers, using the base 10
system, to 100
How does the base 10 system
work?
Why do we use base 10?
causation
number patterns help us to
read and write larger
numbers?
How are these numbers
similar to, and different from,
the numbers up to 10?
The availability of computers
and calculators has provided
us with an unprecedented
opportunity to explore
relationships and rules in the
number system. Students now
have the opportunity to use
technology to find patterns,
explore relationships, and
develop algorithms that are
practical for them. The
educational experiences of
students must include the use
of technology.
Introducing the base 10
system
Students use manipulatives
that can be physically put
together and taken apart, such
as linking cubes. Once
students have understood this
concept, more abstract
manipulatives, such as base
10 blocks, can be used.
Students are given a set of
objects and are asked to write
the numeral indicating the
quantity. Students are given a
number and asked to model
the quantity by using
manipulatives or by drawing
pictures.
Use number–name and
numeral cards to label sets of
objects.
Students need to use
manipulatives and patterns of
number to understand the
concept of the base 10 system.
Students can match a written
number name and a numeral
to a set of objects.
Students need practice in
using manipulatives to
represent a certain number
and writing numbers that have
been represented with
manipulatives.
Students need to practise
reading, writing, comparing
and ordering numbers in a
variety of real-life situations.
Children’s literature also
provides rich opportunities for
developing number concepts.
• count (in 1s, 2s, 5s and
10s), compare and order
numbers to 100
in the numbers to 100?
How can number patterns
help us to count and to learn
more about numbers?
How can we tell if one
number is bigger than
another?
Counting patterns
Students use a calculator to
find number patterns.
They start with zero, add 2,
and record the sum on a 100’s
chart. They continue to add 2
to each sum until they see a
pattern forming. They repeat
the activity by starting with
zero and adding 5 each time,
as well as starting with zero
and adding 10 each time.
When given a larger group of
objects to count, students are
encouraged to group them in
2s, 5s or 10s to help them
count accurately.
How is estimating connected
to counting?
What strategies can we use to
make good estimates?
How can estimating help us to
solve problems?
Estimating to 100
Students are shown a set of 10
cubes. They are then shown
another set of 40 cubes. They
are told how many are in each
set. They are shown another set
of 20 cubes and are asked to
estimate how many cubes are in
this set. Students are asked to
explain the strategies they used
and how knowing the amounts
contained in the other two sets
helped them with their
estimation.
Students can use a 100’s chart
and can look for number
patterns. They can indicate
the patterns found (colouring
in boxes, encircling numbers).
Students can explain how the
number patterns work and can
determine a rule for these
number patterns.
Students need to develop an
understanding of our number
system and the patterns that
arise within it, including the
relationship between numbers,
as well as the properties of
numbers. Students need to use
numbers in many situations in
order to apply their
understanding to new
situations.
Students can group items for
counting and can explain how
this will help.
comparing and ordering
numbers in real-life situations.
Students are asked if there are
more in one pile than in
another and how they know.
This is a great opportunity to
teach the neglected concept of
difference.
Students are shown three
glass jars of the same size and
shape. Each jar contains cubes
of the same size. One jar has
30 cubes, one has 60 and the
other has 90. Students can
estimate how many cubes are
in each of the jars and can
explain the strategies they
used to do this.
Students need a lot of practice
in order to develop their
estimation skills.
They must also have the
opportunity to check their
estimates and to refine them.
Exposure to various
mathematical strategies will
help students to refine their
estimations.
3.30
Content
Number (cont.)
Page 9 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students are given a word
problem and can solve the
problem, write the
mathematical equation and
explain their thinking, using
symbols.
Students need to develop an
understanding that
mathematics is a language that
is communicated through
special mathematical
vocabulary and symbols.
Students will:
• use mathematical
vocabulary and symbols of
add, subtract, difference,
sum, +, –
addition and subtraction to
20 (with and without
regrouping)
How can we record our
mathematical problems?
How can we explain our
mathematical thinking to
others?
How can we model addition
and subtraction problems with
larger numbers?
What equation could help us
to solve that problem?
What problem could help us
to solve that equation?
Using mathematical
vocabulary and symbols
Students are given a picture
showing an exchange of
objects (one person is handing
another person 2 out of 5
apples). Students describe
what is happening by using
mathematical vocabulary.
They record a relevant
equation to describe this
situation.
Add and subtract to 20
Students make up addition
and subtraction word
problems for a partner.
Students play addition and
subtraction games, such as
Snap or Bingo, using different
written and numerical forms.
Students can write an
equation to represent a
problem situation as well as
create a context for an
equation.
Students are given two
problems, one involving
carrying, one involving
decomposition. They can
solve the problem using cubes
and place-value mats.
Students can record the
equations that go with their
models, and can explain how
they solved them.
How do we know when to
regroup? reflection
• automatically recall
addition and subtraction
facts to 10
How can you work out
problems in your head?
How can knowing your
addition facts help you to
solve subtraction facts?
What are good ways to
memorize addition and
subtraction facts?
• describe the meaning and
use of addition and
subtraction
How, and in what ways, do
numbers change when we
add/subtract?
Number facts to 10
Throw two dice. Add (or
subtract) the two numbers,
initially through counting but
building up to automatic
recall.
Students are asked to solve
the problem 4 + 3 in their
head and to explain the
strategy they used. They are
then given the problem 4 + 5
and are asked to solve it using
a different strategy. Students
are asked which strategy was
easier to use and which
allowed them to solve the
problem the fastest.
When do we add or
subtract?
Prepare simple addition and
subtraction problems with the
mathematical symbols missing.
Students are detectives
working out which symbol is
missing and explaining how
they know this.
Students can explain the
strategies they use to solve
problems in their head.
Students can record their
accuracy and speed of
recalling addition and
subtraction facts to 10. They
are asked if they are satisfied
with that speed and what they
could do to become faster.
Students keep a record of
their speed and accuracy over
time to determine
improvement.
Multiple definitions of
subtraction need to be
emphasized, especially “how
many more”, as this leads to
the mathematical concept of
difference.
The use of the appropriate
manipulative materials is
essential for developing an
understanding of regrouping.
Unifix cubes, or another linking
material, need to be used to
introduce these concepts as
students can physically put
these materials together and
take them apart. The use of
more sophisticated materials,
such as base 10 blocks, should
only be introduced once
students have a good
understanding of the concepts.
Students need to practise their
addition and subtraction facts
through a variety of games,
and individual or group
activities, that promote
automaticity. They need to
explore several methods and
discuss these with their peers.
To be useful, addition and
subtraction facts need to be
recalled automatically.
Research clearly indicates that
there are more effective ways
to do this than “drill and kill”.
Above all, it helps to have
strategies for working them
out. Counting on, using
doubles and using 10s are
good strategies, although
students frequently invent
methods that work equally
well for themselves.
Students can describe what
will happen in an addition or
subtraction calculation.
They can explain when they
would use an addition or
subtraction equation.
3.31
Content
Number (cont.)
Page 10 of 10
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
skills
Students can model the
problem and explain the
strategy they used, eg
everyone in the group needs
to have three pieces of paper
for this activity. There are
four people in the group. How
many pieces of paper will
your group need?
Students, at this stage of
development, need to develop
a sense of multiplication and
division as repeated addition
and subtraction. While they
are just beginning to develop
a sense of multiplication and
division through the use of
manipulative materials and
real-life situations, they are
not developmentally ready to
use the mathematical language
or symbols of these operations.
If something is cut in half or
in quarters, students can
explain how many parts it has
and how they know this.
It is important to include
situations where the whole is
a set of objects, as well as one
object that is cut into parts.
Students look at a picture and
can tell the teacher into how
many parts it was cut or
shared.
Initial instruction needs to
emphasize oral language only
and connect it to the models.
Students will:
• explore and model
using their own language/
methods
addition and subtraction help
us?
Is there only one way to solve
this problem?
How does multiplying change
a number?
How does dividing change a
number?
Exploring multiplication
and division
Solve real-life problems
involving multiplication and
division in groups: “There are
20 cubes for this activity and
everyone needs to have the
same number of cubes.”
“There are 5 people in this
group. How many cubes will
each person get?” Students
work with their group to solve
the problem. Each group shares
their strategy and mathematical
thinking with the other groups.
How can we figure out equal
shares?
• use fraction names (half,
quarter) to describe part
and whole relationships
How many parts are there?
Are all these parts the same size?
What can we call the parts?
Halves and quarters
Students cut wholes into parts
to understand their
relationships and to practise
using the language to describe
the parts.
Eight students come to the
front of the room. The class
looks for different ways to
sort these students into
groups.
• estimate the reasonableness
of answers
Does your answer look
reasonable?
Is it bigger or smaller than
you thought it would be?
solving a problem.
What strategies or materials
can you use to solve this
problem?
How can estimation help to
solve this problem?
Can you solve this problem
using a different strategy?
Which strategy was most
effective?
What other problems could
that strategy help us to solve?
Is your answer reasonable?
responsibility
Could that be right?
Students share their ideas,
have a chance to work them
out and check for
reasonableness.
The class can answer questions
about these groupings: What
fraction of the group is male?
What fraction of the group is
female? What fraction of the
group is wearing red? What
fraction of the group has curly
hair?
Students can use their
knowledge of numbers to
estimate an answer.
They can check their answer
against the estimate and
comment on its
reasonableness.
Choosing a sensible method
Students are given a problem
and are asked to solve it using
two different strategies.
Students are asked to explain
which strategies were used
and which was the most
effective.
Students can describe two
strategies for solving addition
and subtraction problems.
Students can explain the
strategy they used, the reason
for choosing that particular
strategy, and the advantages
and disadvantages of each.
Students need to be exposed
to a variety of strategies that
will enable them to construct
meaning about mathematical
operations. They need to
explore various methods of
computation and discuss them
with their peers.
Students need to have
strategies that make sense to
themselves. These may be
their own “invented”
strategies or those that they
have adopted.
3.32
Page 1 of 13
Students will discuss, compare and create sets that have subsets; design a survey; and process and interpret the data on a bar graph where the scale represents larger quantities. They will
manipulate information in a database. They will find, describe and explain the mode in a set of data and will use probability to determine the outcome of mathematically fair and unfair games.
Measurement
Students will estimate, measure, label and compare length, mass, time and temperature using formal methods and standard units of measurement. They will determine appropriate tools and
units of measurement including the use of small units of measurement for precision (cm, mm, ºC). They will also estimate, measure, label and compare perimeter and area, using non-standard
units of measurement. Students will model the addition and subtraction of money and be able to read and write time to the minute and second.
Shape and space
Students will sort, describe and model regular and irregular polygons, including identifying congruency in 2-D shapes. They will combine and transfer 2-D shapes to create another shape.
They will identify lines and axes of reflective and rotational symmetry, understand angles as a measure of rotation and locate features on a grid using coordinates.
Students will recognize, describe and analyse patterns in number systems. They will identify patterns and rules for multiplication and division, together with their relationship with addition
and subtraction. They will model multiplication as an array and use number patterns to solve problems.
Number
Students will read, write, estimate, count, compare and order numbers to 1000, extending understanding of the base 10 system to the thousands. They will read, write and model multiplication
and division problems. They will use and describe multiple strategies to solve addition, subtraction, multiplication and division problems, reasonably estimating the answers. They will
compare fractions using manipulatives, mathematical vocabulary and fractional notation. They will understand and model the concept of equivalence to one.
Content
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• discuss, compare and create
sets from data that has
subsets using tree, Carroll,
Venn and other diagrams
What is the relationship
between these two sets?
What smaller set does this set
contain?
Discussing and creating sets
Students sort drawings or
names of objects, rather than
the objects themselves, using
a tree diagram.
Students investigate how
many different combinations
can be created from a set of
leisure clothes. The concept of
a tree diagram is introduced
and is used to show a
systematic way of accounting
for all possibilities.
When and why would we use
these diagrams?
• design a survey, process
and interpret the data
How can we find out about
this?
What are the most important
pieces of information we need?
In what ways can we display
the data collected?
How can we interpret the data?
Who might be interested in,
or be able to use, the results
of our survey? perspective
Use the same information
(teacher-generated and given
to the students or collected by
the students) and display it on
a Carroll diagram and a Venn
diagram.
Designing a survey
Students choose a topic and
frame questions appropriate
for a small survey. Before
conducting the survey, they
discuss the best way to obtain
the information: who/how
many people will be asked,
clear wording for questions,
recording methods. They
conduct the survey, collate
and represent the data using a
variety of forms, and discuss
the findings.
Given a specific set, such as
names in the class, students
suggest labels for paths/
branches of a tree diagram.
Natural examples of subsets
can be explored within the
units of inquiry.
Students can draw a diagram
to show the names of known
coins and whether they are
copper- or silver-coloured.
Students can draw a Venn
diagram to show the sets
“students in our school”,
“students in our class” and
“students in our group.”
Students can highlight areas
where the information
corresponds in the two
formats.
A great deal of practice may be
necessary for some students to
describe the two (or more)
attributes often shown in
Carroll and Venn diagrams.
Students can carry out a
survey to collect and collate
information about classmates.
Students can represent the
data and are able to explain
any results or patterns.
Presenting data
diagrammatically often helps
understanding or analysis and
facilitates explaining data to
others.
It is important to remember
that the chosen format should
without bias.
What could you do differently
if you repeated the survey?
reflection
3.33
Content
(cont.)
Page 2 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• collect and display data in a
bar graph and interpret
results
What are the features of bar
graphs?
What does this bar graph tell
us? function
How do these features
compare with other graphs
you have made?
How is a bar graph like a
pictograph?
How are they different?
• use the scale on the vertical
axis of a bar graph to
represent large quantities
How can we fit data on this
piece of squared paper when
the range of numbers is so
large?
On what should we base our
decision?
• find, describe and explain
the mode in a set of data
and its use
Which is the most common
eye colour?
Why would finding the most
common be important?
What do mathematicians call
this?
How do we want to collect
the data?
How will we record and
organize the data?
• understand the purpose of a
database by manipulating
the data to answer questions
and solve problems.
How does a database work?
How does it organize the
information?
What do we need to know in
order to operate a database?
When is it most appropriate to
use a database?
Given a set of data, what
different searches could we
carry out?
Bar graphs
Teachers could illustrate to
students by using Cuisenaire
rods and then exchanging
these for the matching rod
representing the total number
in the set: a yellow rod would
replace five white unit rods.
Students name the TV
programme they watch at a
given time. Data is
represented using interlocking
blocks. Each column is then
labelled. The data could be
transferred to paper.
Making the data fit the
graph
Use scales where each unit is
not recorded but marks are
indicated in intervals of 2 or
5. Use squared paper to give
easily marked intervals.
When a frequency table is
being prepared, introduce
students to a tally system to
make the total easier to count.
What is the mode?
Surveys are carried out to find
the mode for the class/school
of height, weight, amount of
allowance, number of
siblings, favourite cartoon,
favourite singer/band,
favourite sports team,
nationalities, ages.
Using the mode as a guide,
students produce a shirt that
will fit most of the class, and
make a paper pattern for it.
When do we use a
database?
naturally in the classroom or
form part of the units of
inquiry present opportunities
for students to use databases.
Any collection of data can be
sorted and classified.
Visit a local library that uses a
database to organize and
maintain information about
the books it collects. Discuss
with librarians and assistants,
and compare with previous
organizational systems.
Students can draw a bar
graph to display given data.
Students can collect their own
data and display it in a bar
graph.
students can begin to use bar
graphs to record data because
they can be simply viewed as
putting the squares or tiles
together in a rectangle.
Students can describe and
discuss key features of the
graph.
They should be given
opportunities to do this almost
on a daily basis. The units of
inquiry will also be rich with
opportunities for graphing.
Students are given a data set
and possible scales. They can
determine and defend an
agreeable scale.
naturally in the classroom or
form part of the units of
inquiry present opportunities
for students to determine a
scale. Students need to be able
to suggest possible solutions,
put them forward for
discussion, and defend or
adapt their choice.
As students deal with larger
data samples, they can solve
the problem of how to fit all
the data on one piece of
paper.
When given a set of data,
students can determine the
mode and explain why
knowing the mode is useful in
this situation.
Students can describe in their
own words what the mode
value is (oral).
Using data that has been
collected and saved is a
simple way to begin
discussing the mode. A further
extension of mode is to
formulate theories about why
a certain choice is the mode.
They can find the mode for
the number of sweets in
several bags (practical).
Students can develop
questions they want to answer
given a particular database.
This demonstrates knowledge
of fields. They can use the
database to answer their
questions and present findings
to the class.
Students should have the
opportunity to use prepared
databases, either those that
are commercially made or,
ideally, those created using
data collected by the students
then entered into a database
by the teacher/together.
The focus here should be on
using a database, not on
creating one. Often new and
unsuspected relationships will
appear.
Where else in school or life
do we use databases?
connection
3.34
Content
There are ways of finding out
if some outcomes are more
unlikely, certain or impossible.
It can be expressed
quantitatively on a numerical
scale.
Page 3 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• use probability to
determine mathematically
fair and unfair games and to
explain possible outcomes.
What makes a game fair?
Is this game fair to all
players?
How can we tell?
How can we change the game
to make it more fair?
opportunity to process data
and explore probability in
more thoughtful, efficient and
imaginative ways. The
of technology.
Have we tested enough?
responsibility
Is it fair?
Assign the numbers 1 to 12,
to represent the sum of two
dice, to students in the class.
Predict which student will win
when the dice are rolled 50
times. Then make a systematic
list of all the possible
outcomes.
Students can invent a game.
They can design two sets of
rules, one that is fair and
another that is unfair. They
can explain why this is the
case.
Students can count how many
ways there are of making each
sum and can explain if this
game is fair.
Students record the outcome
of two coins being tossed. If
identical coins are tossed, one
at a time, the sequence of
outcomes could be recorded
using words or symbols: HH,
TH, HT, TT. If the coins are
tossed together, outcomes for
the pair should be recorded as
“two heads”, “a head and a
tail”, “two tails”. Students
make predictions about the
number of heads and/or tails
after 10 trials.
Students can name possible
outcomes (two heads and a
tail, three heads) when three
coins are tossed at the same
time. They can repeat the
experiment to show that some
outcomes are much less likely
than others.
Students use two spinners
each with the numerals 1 to 4
equally likely to land.
Students can discuss the
possible totals. They can
match predictions against
actual outcomes.
Our sense of probability often
defies mathematical
definitions and should not be
dismissed. A good example is
flipping a coin. Although
every time you flip a coin, the
mathematical chance that it
will fall on one side rather
than the other is equal, many
people would say that after a
run of one side, the chance of
flipping the other side
increases. There is some
common sense in this.
Discussion is needed to
explain how the spinners are
designed to ensure a fair
game.
Discussion of results should
be related to the number of
trials.
Measurement
are continuous, ways must be
found to deal with quantities
that fall between the numbers.
be or can ever be.
and compare using formal
methods and standard units
of measurement: length,
mass, time and temperature
How can we measure the size
of something?
How can we measure the
amount of space it takes up?
Which of these rocks is the
heaviest?
Measuring using standard
units
Use base 10 blocks to show
that there are 100cm in 1m.
Students compare the number
of paces across the
playground with a
measurement using a 1m
ruler.
Students are given a piece of
streamer or string 1m long
and can estimate which
objects in the classroom are
closest to 1m in length.
Measurements are then
checked.
Students use 1m rulers to
measure larger objects in the
classroom: desks, window
ledges. They decide what
whole number of metres the
measurement length is closest
to. Rounding off rather than
recording in metres and
centimetres is important.
Students can estimate a given
distance in metres and can use
their measuring tools to
check.
Students use a thermometer to
record temperatures in
different parts of the school at
different times of the day.
Students can prepare a map of
the school showing hot or
cool spots and other spots
where the temperatures are
likely to change during the
day.
Students use a 1kg mass and a
pan balance to sort a variety
of objects from home or the
classroom that will weigh
exactly 1kg, more than 1kg
and less than 1kg.
Informal and personal
systems of measurement need
to become standardized so we
can communicate
measurement accurately.
The process of measuring is
identical for any attribute:
choose the unit, compare that
unit to the object and report
the number of units.
A trundle wheel could be
introduced to check distances.
An estimation followed by a
measurement is better than
doing many estimates and
then checking the estimates.
Students need a chance to
refine their estimates on the
basis of feedback. Having
students discuss a range of
acceptable estimates, and how
they determined their own
estimate, helps everyone
develop a strategy that makes
sense and is useful.
3.35
Content
Measurement (cont.)
Page 4 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students can select an
appropriate measuring tool
from a variety available
(ruler, tape measure, metre
rule, trundle wheel) to
measure the length of a book.
A wide range of measuring
tools should be available to
the students: 30cm rulers,
metre sticks, 30m tapes,
trundle wheels, height
measure charts, tape
measures, bathroom scales,
pan balances, kitchen scales,
masses (1kg, 500g, 100g),
sand timers, egg timers,
analogue clocks, digital clocks,
stopwatches, calendars.
Students will:
• select appropriate tools and
units of measurement
What tools do mathematicians
use to measure length, mass,
time and temperature?
What are we trying to
measure?
Which tool is best for this
measurement?
Can you suggest any other
ways of finding this
measurement?
How can we measure this?
Students measure a variety of
objects: short and long
lengths, light and heavy
masses, hot and cold
temperatures and so on.
Before measuring, they must
decide upon the attribute they
wish to measure. They select
the tools they think will best
measure that attribute. After
measuring, students discuss
whether another tool, or
choice of unit, would have
worked better.
What units should we use?
What sort of objects could we
measure with smaller units?
How do we read the smaller
units?
How accurate does the
measurement have to be?
responsibility
Give the students
opportunities to measure the
same attribute using units of
different sizes (mm and cm).
This will help them to select
the most appropriate unit.
Students, individually or in
pairs, can choose the most
appropriate unit for measuring
the length, height or width of
a designated item from a list.
Each student measures, in
centimetres, a feature (height,
width, length) of a portable
object in the classroom. Results
are recorded on cards.
Students can match each card
with the object measured.
Students estimate, in
centimetres, the lengths/widths
of their index fingers/feet.
Exact measurements are then
taken and results checked.
Students research a sport of
interest to them and discuss
the importance of precise
measurements in that sport
(eg Olympic records).
• describe measures that fall
between numbers on a
measure scale: 3½kg,
between 4cm and 5cm
What do we do when the
object we are measuring falls
between two whole numbers?
How can we describe a
measure that falls between
whole numbers on the scale?
Which parts of the tools help
us?
What do we call the units
between the numbers on the
scale?
and compare perimeter and
area
What is perimeter? form
What is area?
How is area connected to
perimeter? connection
What happens when we
change from one unit to
another? change
What did we use to measure
with and why did we choose
it?
Measures that are not exact
From a supermarket catalogue,
students select up to 10 objects
whose mass is given in grams
and kilograms. They sort the
items into categories, eg those
within 100g of 1kg, and those
that are between 800g and
1.5kg.
Students are asked to choose
an object that is between 5cm
and 6cm long. “How we can
describe the size.”
Students naturally find
solutions for measures that
fall between two units: “It’s
more than 100g”, “It’s
between 11cm and 12cm.”
Students collect maximum
and minimum daily
temperatures from
newspapers, or the web, for
capital or regional cities
around their home country.
Students look for patterns
over time.
Looking at area and
perimeter
Students are given real objects
and situations in which they
must compare the area and
perimeter of objects.
Students choose things in
their immediate environment
to measure area. They discuss
why each item chosen may or
may not be appropriate. They
use these things to measure
area.
Give the students real objects
in situations where they must
measure items. These
situations should come from a
unit of inquiry in order to
make the use of measurement
more authentic.
Students can find the
perimeter and area of these
shapes using non-standard
units of measurement.
Perimeter is the length of the
outline of a shape or region.
The units are, therefore, the
same as for length.
Students can measure
different perimeters (length
units) for the same area.
Area is the amount of space a
shape covers and is measured
in square units.
Students can use non-standard
units of measurement to
estimate and measure the
areas of a variety of shapes in
the classroom.
Possible non-standard units of
measurement include fingers,
hands, books, tiles, base 10
flats and units, shoes, erasers
and paper clips.
3.36
Content
Measurement (cont.)
Page 5 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
many, of the specific expectations
and transdisciplinary skills
On cm2 paper, draw three
different rectangles with the
same perimeter. Which
rectangle has the greatest
area? Students can
demonstrate the ability to
draw three rectangles
correctly, to determine
perimeter, to determine area
by counting square units, and
to identify the rectangle with
the greatest area.
When students have
demonstrated an
understanding of perimeter
and area and how they can be
measured, standard units of
measurement can be
introduced.
Students can find out how
much it will cost to buy their
favourite comic/magazine for
one year.
Use place value mats/boards,
bills/paper money and
decimal bars or coins, and
counters.
Students will:
How should we deal with any
remaining parts?
How many different shapes
can be made that have the
same area?
Are our estimates realistic?
reflection
Looking at area and
perimeter (cont.)
Students use non-standard
units of measurement to
measure and compare two
regular or irregular shapes.
Students discuss which shape
has the larger area and how
this relates to the unit chosen.
Standard units of measurement
can then be introduced.
Students use square tiles to
make shapes with the same
area but different perimeters.
Students use geoboards to
investigate how different
shapes can have the same area.
• model addition and
subtraction using money
How could the tenths and
hundredths on a place value
mat/board relate to money?
connection
Addition and subtraction
with money
Students play trading games,
using play money, with an
emphasis on the base 10
system.
Each student is allowed a set
amount of money (50 dollars,
pounds, euros etc). They buy
items from a catalogue and
keep a running total. They try
to come as close as possible
to spending the money
without going over.
• read and write the time to
the minute and second,
using intervals of 10
minutes, 5 minutes and 1
minute, on 12-hour and 24hour clocks.
What are minutes? form
What are seconds? form
How can we record time?
function
How long do you think it will
take you to…?
How long did you spend
doing…?
If a movie begins at 4.15 and
ends at quarter past six, how
long did it last?
What do you think the saying
“time flies when you’re
having fun” means?
reflection
Telling the time
Students discuss different
units for measuring time
(seconds, minutes, hours,
days, weeks) in association
with familiar experiences
(coming to school).
Students record various
activities that they estimate
could be completed in about a
minute, an hour, 5 minutes,
half an hour.
Students time various
activities in their day using a
stopwatch and estimating at
first. They then record how
long it will/did take.
The teacher sets out a given
amount, then students add or
subtract from it as much as
requested and can give change
in a shopping transaction.
Letters of the alphabet are
assigned a monetary value.
Students can use these to
calculate the value of names,
words and sentences.
A younger sibling is learning
to tell the time on a digital
clock and wants to know the
meaning of “am” and “pm”.
How can you explain it?
(Students can give a logical
explanation.)
Students, alone or in groups,
can identify the most
appropriate units of time to
measure the duration of a
range of different events: a
school play, a movie, a soccer
match, feeding the class pets.
Time lines from noon to noon
or midnight to midnight,
where 12-hour and 24-hour
notation is recorded, can be
used.
This is a good opportunity to
use key vocabulary: duration,
analogue, digital.
Count in 5s to tell the time to
the nearest 5 minutes and
report it orally: 4 o’clock and
20 minutes.
Match the time on a digital
clock to the time on an
analogue clock.
Match the time in 12-hour
notation to 24-hour notation.
3.37
Content
Shape and space
Page 6 of 13
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Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students can use geoboards to
make a seven-sided polygon
with at least one right angle.
Students need to see and
develop their own conceptual
framework for polygons based
on positive and negative
examples of a variety of
polygons.
Students will:
• sort, describe and model
regular and irregular
polygons: triangles,
hexagons, trapeziums
What polygons can we see
around the classroom?
What are the properties of
polygons?
Exploring polygons
Flat shapes are used to
explore similarities and
differences: finding all the
shapes with only straight
sides, with more than four
corners.
How are these shapes
similar/different?
Students can devise a
checklist to help sort 2-D
shapes by their properties.
Students can describe how
these two polygons are similar
or different.
What do these shapes have in
common?
Both regular and irregular
examples of polygons should
be investigated.
How could we sort these?
How could we sort them in a
different way?
Students can explain why a
triangle is a polygon and a
line is not.
Which rule do you think
he/she used to sort the
shapes?
• identify, describe and
model congruency in 2-D
shapes
Can you find two identical
shapes?
What makes them identical?
What is congruency?
Students use a computerdrawing package and several
techniques to build congruent
figures.
Students can identify items
that are congruent and explain
how they know this.
Students need to have many
experiences with shapes that
are congruent. They need to
practise describing what
makes them congruent.
When will our knowledge of
shapes be useful?
responsibility
What happens if we flip or
turn the shape? Is it still
congruent? (link to symmetry)
• combine and transform 2-D
shapes to make another
shape
How can 2-D shapes be
combined?
combine 2-D shapes?
What new shapes can we
make from these shapes?
stretch the sides of a shape?
change the angles on a 2-D
shape?
What name do
mathematicians give to these
new shapes?
Geoboards and geostrips are
excellent materials to use in
these activities.
Through modelling and
manipulating shapes, students
of shape with more formal
begin to appreciate the need
for this precision.
Combining 2-D shapes
Using geoboards or geostrips,
students transform 2-D shapes
by asking questions: What
happens to a square when you
make the sides longer? What
happens to a rectangle when
you change the angles?
Students, using various
materials, move, combine and
manipulate shapes to make
other shapes.
Given two shapes, students
can combine them to make
two new shapes.
Given an original shape and
its transformation, students
can describe what was done to
the shape.
experiences combining shapes
through building, drawing and
construction.
By understanding how one
shape can be transformed into
another (a square into a
rectangle) students can begin
to recognize and verbalize the
properties of each.
Students pin a shape template
to paper and explore the
shapes made when they
continually rotate and trace
the shape.
3.38
Content
Page 7 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• create symmetrical
patterns, including
tessellation
In what ways are the patterns
similar/different?
Where in the designs do we
see a flip (reflection), turn
(rotation), slide (translations)?
What makes certain designs
work?
What can you say about the
area covered by each of our
tiling patterns?
Where else have you seen
patterns like the tiling
patterns?
Does a…tessellate? function
Do all quadrilaterals
tessellate? Why? causation
• identify lines and axes of
reflective and rotational
symmetry
What is an axis of symmetry?
form
How does it help us define if
things are symmetrical?
What happens if you place the
edge of a shape against the
edge of a mirror?
What happens if we move the
mirror further and further away
from the edge of a shape?
Which of these shapes do you
think you could fold so that
the two sections match up?
Creating symmetrical
patterns
Create a school logo using
rotational symmetry.
Tiling patterns.
Describe the pattern of tiles
on the classroom floor.
Using only one kind of block,
find which can be used to tile
or cover an A5 size piece of
paper. Then, using only two
or three different blocks, try to
create a tile pattern on the paper.
How can we measure the size
of a turn that we make?
If we turn from north to
south, how much of a turn do
we make? North to east?
How do mathematicians
describe this?
How can we compare and
describe these rotations?
Art connection: use shapes to
create tessellating mosaic
patterns and designs.
Students can make a template
of a tessellating shape. They
can use this to create a pattern
and to write a description of
the pattern.
tile their table tops, using
multiple copies of one shape,
and several colours, to create
a pattern.
Make as many groups of
squares as possible. Each
square must share a full side
with at least one other square.
Wherever the squares touch,
it must be with full sides.
Identifying lines of symmetry
Students take a survey and
make a graph to show the
most common shape in the
classroom and to identify
objects in the classroom that
have reflectional symmetry.
Students use one 2-D shape to
make three patterns: one by
flipping, one by turning, and
one by sliding the shape.
Use geoboards to show
shapes that have symmetry on
each side of an imaginary
mirror line and those shapes
that do not.
What happens if you place the
mirror on the fold line?
• understand an angle as a
measure of rotation by
comparing and describing
rotations: whole turn; half
turn; quarter turn; north,
south, east and west on a
compass
When students are shown a
tiling design they can
determine where flips, slides
and turns are used in the
design.
Angles as a measure of
rotation
Students devise and play a
game that uses compass
directions and turns as part of
the rules.
Students practise moving
themselves, robots and objects
through space and keeping
track of paths in order to
describe them to someone else.
Use a clock face, with moving
hands, to ask the time if the
minute hand makes a quarter
turn. Ask students to justify
their responses.
Students can find lines and
axes of symmetry and explain
how they know this.
Students can sort a range of
regular and irregular 2-D
shapes into those with
symmetry and those that are
asymmetrical. They can draw
the shapes showing any lines of
symmetry.
Students can write riddles
describing the features of a
secret shape.
Students can make the mirror
image or reflection of an object
by placing the mirror line in
several different positions.
Students can give three sets of
directions to move an object
from a starting position to a
new position.
When given a route map of a
robot, ship or person’s
movements, students can
describe the visual image as a
series of turns and can
estimate the size of the turns.
experiences folding paper,
using mirrors and rotating
objects to find lines of
symmetry.
Symmetry is an important way
of understanding and
comparing shapes.
Students need to experience
both reflective and rotational
symmetry.
Students could use computer
software to explore the effect
of flip and rotate tools on a
complex shape or picture.
With computer packages and
programmable toys, students
can become familiar with
turns of 90°, 180° and 360°.
Practical activities with
quarter, half and whole turns
help connect this to students’
personal experiences.
Note the link to the
measurement expectation for 9–
12 years.
3.39
Subject: Mathematics
Content
Age Range: 7–9 Years
Page 8 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
After having set up their own
grid system, students can tell
other students where to place
an item. A fellow student will
place the item in the correct
position.
A unit of inquiry in which
students use maps is an
excellent way to practise this
skill and to explore different
methods of arranging a grid.
Students will:
• locate features on a grid
using coordinates.
position of something?
Locating features on a grid
Students put their desks in
columns and rows to form a
grid. They discuss ways to
identify each desk by its
position on the grid.
Play a coordinate game using a
10 × 10 grid where the x’s and
o’s are put on the line
intersection instead of in
spaces. They are placed
according to their ordered pair
names (eg 4, 2 where 4 refers
to the horizontal axis and 2
refers to the vertical axis). The
goal is to get 4 x’s or 4 o’s in a
row.
Students can draw a map
showing two paths from their
classroom to the fire drill
area. They can write
accompanying directions.
Students can use a local map
to write and answer location
questions: What is at C3?
Students select a mystery
object in class and describe its
position accurately. Others
guess the object.
to understand how
of algebra.
Students will:
• analyse patterns in number
systems to 100
• recognize, describe and
extend more complex
patterns in numbers
What patterns can you find in
the 100s chart?
What patterns do you see?
What are the features of this
pattern?
What will be the next item in
the pattern?
How can we move the
numbers around to get a
different pattern?
How can we describe this
pattern verbally, in a table, in
a formula, in a graph?
Analysing number patterns
Students use a 100s chart to
code a variety of patterns:
numbers that have 5s in them,
numbers that end in zero.
They then describe the
patterns that emerge.
More complex number
patterns
Calendar patterns.
Jumps and Slides game
(9 spaces, 4 blue cubes, 4 red
cubes – 4 of any two different
things), where the goal is for
the colours to exchange places
by moving one space at a time;
each colour can jump over the
other colour but not over its
own.
RRRR
BBBB
Given a 100s chart with a
missing number, students can
correctly place the number
and justify their answer.
They can also describe and
explain a pattern they find for
themselves in the 100s chart.
Students will continue to use
patterns as they progressively
learn the number system.
Use these five numbers to
make a pattern: 11, 9, 14, 7,
12. Give the rule for your
pattern. One possible
response is 7, 12, 9, 14, 11.
The rule is add 5, subtract 4.
Patterns are central to the
understanding of all concepts
in mathematics. They are the
basis of how our number
system is organized.
Searching for, and
identifying, patterns helps us
to see relationships, make
generalizations, and is a
powerful strategy for problem
solving.
Students can solve the
following problem: How
could I find how much money
I have if I have seven piles of
coins each containing 5 one
cent/pence coins?
Colours can slide across but
only if there is a free space next
to them; colours can only move
in the one direction, never go
back. This problem works well
if you use the “make it easier”
strategy: instead of four of each
colour, start with one, then do
two of each etc, until you see
the pattern.
Students input a secret
counting pattern into a
calculator. Others can use the
constant function to repeat the
pattern and guess the rule.
What will be next in the
pattern? Why?
Show a pattern with attribute
blocks. Ask students to
continue the pattern with
partners, taking turns to add
an attribute block.
Students can say what could
be next and why. There may
be several possible responses
but each one changes in only
one way.
What patterns are in a row of
a multiplication table?
Students explore patterns for
each multiplication table.
Students can explain what 28,
35 and 42 have in common.
What do you think a
“growing” pattern would look
like?
Functions develop from the
study of patterns and make it
possible to predict in math
problems.
Use pattern blocks, attribute
blocks, colour tiles,
calculators, number charts,
markers of all types (beans,
buttons), spinners.
How can patterns help us
remember our multiplication
facts?
3.40
Content
Page 9 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students can explain how 5 +
3 = 8 and 8 – 3 = 5 are
connected. They can make a
model to show this.
Besides introducing the
important mathematical
concept of inverse operations
in a user-friendly context,
finding families of number
facts is another way for
students to learn and
remember the subtraction
facts.
Students can explain how a
multiplication or division
pattern will continue, using a
number chart or manipulatives.
Commutative and associative
properties apply to
multiplication and addition
but not to division or
subtraction.
Students will:
• understand and use the
relationship between
4 + 3 = 7, 7 – 3 = 4
subtraction connected?
When might understanding
this be helpful?
Using the relationship
between addition and
subtraction
Using real-life problems,
students experience the idea
that if 3 apples + 4 apples = 7
apples, then if we have 7
apples and eat 3 of them, we
will be left with 4 apples.
How does the way in which
numbers are grouped affect
the answer?
for multiplication and
division:
4 % 3 = 12, 3 % 4 = 12,
12 + 3 = 4, 12 + 4 = 3
How are multiplication and
division connected?
Why do these calculations
produce patterns? causation
How does the way in which
numbers are grouped affect
the product? (commutative
and associative property)
Where else in mathematics
does this rule apply?
connection
(repeated addition)
division connected?
connection
How can our calculators
count by 10?
Using a 100s chart, colour in
every 3rd, 4th and 5th square to
visually demonstrate patterns.
Draw circular patterns on
clock faces, where every 3rd,
4thand 5th number is joined
with a line to demonstrate
patterns.
Students are taught to identify
the rule that products are
always larger than factors,
and that quotients are always
smaller than dividends.
division related?
Families of numbers are
investigated to demonstrate
the connection between
multiplication and division:
3 % 7 = 21
7 % 3 = 21
21 + 7 = 3
21 + 3 = 7
addition related?
Students work in pairs, using
calculators to complete charts.
They press the keys
then
2
One person presses
=
+
Suppose that this
mathematical sentence
represents a division fact:
divided by
=
Students can write the other
facts from this family.
Given a number to count by,
students can predict the
pattern of counting the first
three entries.
Use four-function calculators.
then presses it again as they
watch the pattern grow.
Students are asked how they
can use a calculator to find the
product of 6 % 27 if the %
button is broken.
How are division and
Students work in pairs to
identify patterns by
repeatedly subtracting a
number.
Students are asked how they
can use a calculator to find the
quotient of 96 + 4 if the +
key is broken.
Students can calculate the
product without using the
key.
%
quotient without using the
+ key.
Care should be taken to help
students choose an
appropriate starting number
(dividend) and divisor to
avoid remainders.
3.41
Content
Page 10 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• model multiplication as an
array
How can we model a
multiplication equation?
Which products can be
represented as squares and
which as rectangles?
Arrays
Students use arrays to explore
multiplication facts and the
relationship between various
multiplication facts.
Students make models of all
the possible rectangles with
an area of 1 to 25, using tiles.
Students use counters to
explore making square and
rectangular numbers.
Students create rectangular
arrays that have equal areas
but different dimensions.
Progress from making arrays
with tiles to using graph
paper.
• understand and use number
patterns to solve problems
(missing numbers).
What symbols do
mathematicians use to
represent unknown numbers?
Missing numbers
Students practise with a function
machine to understand that a
function applies a consistent rule
when it operates on a number.
Students continue the pattern
to build a row of 100
triangles: How many
toothpicks are needed?
1 triangle needs 3 toothpicks,
2 triangles need 5 toothpicks,
3 triangles need __ toothpicks.
Try this with squares,
pentagons, hexagons.
Students can model 2 % 3 and
3 % 2 and can explain how
they are the same and how
they are different.
Students can explain that the
area of a rectangle can be
described as a multiplication
problem.
Students can describe all the
possible dimensions of a
regular-shaped garden if its
area is 24 square units.
Besides deepening their
understanding of
multiplication, students will
discover the commutative
property of multiplication,
which will help them in
learning the multiplication
facts.
The resulting records of these
rectangles form the basis of
discussion about the square
numbers, prime numbers,
multiples and factors.
They can calculate the area of
a 8m by 6m garden lawn,
using an array.
Students can identify the rule
or function being used in the
following equation:
5 + ___= 6 and 7 + ___= 8.
Students can identify missing
elements of number patterns
and can discuss attributes of
each pattern.
Generalizing a pattern and
calling it a function is a
central idea in mathematics.
Students need practice in the
formation of functions and in
determining functions from a
t-chart (graph) in order to
develop algebraic thinking.
Suppose everyone in the room
shakes hands with every other
person in the room. How
many handshakes will that
be?
Number
on its place within a base 10
The degree of precision
needed in calculating depends
on how the result will be
used. The availability of
computers and calculators has
provided us with an
unprecedented opportunity to
explore relationships and
rules in the number system.
system, to 1000
How do mathematicians write
larger numbers?
Why do we use base 10?
causation
How can we use the base 10
system to show large
numbers?
Using numbers to 1000
Students represent 1000 in a
variety of ways.
Students practise using base 10
blocks to represent a certain
number and practise writing
numbers that have been
represented by base 10 blocks.
Students use base 10 materials
to play trading games with an
emphasis on base 10 structure
and place value.
Students can explain how
they know what number this
is.
Use place value mats/boards
to extend the base 10 system,
step by step, to the next place.
Students are given the
following scenario. A new
student comes to their class
who has missed this unit on
place value. They can explain
to the new student how the
base 10 system works.
Students play a calculator game
that focuses on place value
understanding (eg 4293: how can
you change the digit 4 to a 9?)
• count, compare and order
numbers to 1000
Which digit is the greatest?
How do we know?
Students use base 10 materials
to play trading games with an
emphasis on base 10 structure
and place value.
3.42
Content
Number (cont.)
Students now have the
opportunity to use technology
to find patterns, explore
relationships and develop
algorithms that are practical
for them. The educational
experiences of students must
include the use of technology.
Page 11 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
in a variety of ways..
Students will:
How is estimating connected
to counting?
• count in 3s, 4s, 5s, and
explore other numbers
When would it be more
efficient to count by 3s, 4s…?
Estimating to 1000
Students are given
opportunities to estimate and
to check their estimates.
Counting in groups
Students group manipulatives
in 3s, 4s or 5s and count totals.
They discuss which method
was the easiest and why.
Explore counting patterns
using 100s charts and
calculators.
• use number patterns to
learn multiplication tables:
1s, 2s, 4s, 5s, 10s
What number patterns do you
know?
Learning multiplication
tables
Students use arrays to explore
multiplication facts and the
relationships between them.
When do you need to use
multiplication tables?
• automatically recall basic
facts
What strategies do you use to
help you learn the basic facts?
memorize the addition facts?
memorize the subtraction
facts?
Addition and subtraction
facts
Students practise their facts in
a variety of ways that promote
correct recall: games,
individual and group activities.
Students use their
understanding of addition
facts and strategies to work
out the subtraction facts.
Students explain and compare
the strategies they use to solve
mental computations.
subtraction equations to
1000 (with and without
regrouping)
to solve this problem?
What problem could this
equation help us to solve?
Adding and subtracting to
1000
Students use base 10 blocks to
represent addition and
subtraction (to the 100s place)
and, given a representation,
write the equation represented
by the blocks.
they estimate, how many
things they think are there,
how they know, and how they
think they might check their
estimate.
Find authentic contexts in
your units of inquiry where
estimating can be practised.
Students can input a secret
counting pattern into a
calculator. Others can use the
constant function to repeat the
patterns and to guess the rule.
Students can predict,
demonstrate and explain
various counting patterns.
Students can recognize and
describe number patterns they
see. They can understand to
which multiplication table
they are linked.
Research continues to indicate
that “drill and kill” methods
are only slightly effective and
that there are better ways to
practise these skills. It helps
to have practised, learned and
discussed strategies; students
can then call upon these to
help work out facts.
Ask students how fast they
can recall the basic facts and
if they are satisfied with that
speed. Help them devise a
way to time themselves.
Discuss ways of improving
their speed if they are
interested.
To be useful, basic addition
and subtraction facts need to be
recalled automatically.
Students can mentally solve
computation problems for each
operation and are able to write
the strategies used for each.
Counting on, using doubles and
using tens are good strategies,
although students frequently
invent methods that work
equally well for themselves. To
increase the speed of recall,
most students respond best to
graphing their speed over time
on a small set of similar facts.
Students can calculate
equations, using
manipulatives, from written
and numerical problems.
Why do we regroup?
causation
multiplication and division:
times, divide product,
quotient, %, +
What language is used when
we multiply or divide?
What symbols do
mathematicians use to show a
multiplication or division
equation?
Using mathematical
vocabulary and symbols
Students explain their work to
each other using new
vocabulary and symbolic
representation.
Students can correctly use
symbols when calculating
multiplication or division.
What is a product?
What is a quotient?
3.43
Content
Number (cont.)
Page 12 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students can describe two
strategies for solving addition
and subtraction problems, and
the advantages and
disadvantages of each.
Students should be exposed to
a variety of strategies that will
enable them to construct
meaning about basic facts.
They need to explore several
methods and have
opportunities to discuss them
with their peers.
Students will:
• use and describe multiple
strategies to solve addition,
subtraction, multiplication
and division problems
How can we solve this
equation?
Using multiple strategies
Students share their calculation
methods with the class.
What mathematical operation
will help us to solve this
equation?
Students can write a +/equation to represent a
problem situation, as well as
create a context for an
equation.
Are there other ways to solve
the problem?
answer need to be?
How is two- and three-digit
addition or subtraction the
same/different when you
work in your head, with paper
and pencil, or with a
calculator? perspective
problems
What equations could we use
to solve this problem?
Calculator skills must not be
ignored. All answers should be
checked for their
reasonableness.
Multiplication and division
strategies
Students calculate the cost of
buying the same snack/lunch
every day for various periods
of time.
Students calculate the total
cost of a class field trip when
given the cost per student.
Students, given the total cost
and the number of students
attending, calculate the cost
per head of a class field trip.
• compare fractions using
manipulatives and using
fractional notation
Can different fractions be
equal?
How can we know when one
fraction is greater than,
smaller than or equal to
another?
The research is unequivocal:
people invent or adopt
methods that make sense to
them. The methods they use
are varied, flexible and often
lead to good estimation skills.
Comparing fractions
Students use fraction pieces to
compare fractions and to
record the comparison in an
equation or an inequality.
Students make their own
fraction strip kits using
different coloured paper.
Students match cards using
fraction words, diagrams and
notation.
Students divide the clock face
into a range of common
fractions and relate these to
parts of an hour.
Students practise using
fraction circles and fraction
pieces to represent a given
fraction, and practise writing
fractions that have been
represented with materials.
Students are given a range of
numbers. They can use
several of these to write and
solve problems involving a
variety of mathematical
operations.
Given a multiplication
equation such as 17 % 9,
students can build or draw a
model of that operation.
Given a visual representation
of a multiplication situation,
students can write the
equation.
Students can find and explain
examples of common
fractions and objects: sport
courts, window panes, clock
faces, egg cartons.
Students are asked if the
following inequalities are true
and can explain their
thinking:
2
/3 > 3/4, 2/4 > 4/8.
Students can explain what 2/4
is the same as.
Given a fraction, students can
build or draw a model of the
fraction. Given a model,
students can write the fraction
using fractional notation.
They can construct a whole
when given a part.
The interpretation and
meaning of remainders can
cause difficulty for some
students. This is especially
true if calculators are being
used. For example, 67 + 4 =
16.75. This can also be shown
as 16 ¾ or 16 r3.
Students need practice in
producing appropriate answers
when using remainders. On a
school trip with 25 students,
and buses that carry 20
students, then a remainder
could not be left behind –
another bus would be required!
Difficulties with fractions can
arise when fractional notation
is introduced.
Language is important.
“Three out of five” is a
helpful way to describe 3/5.
Students move from a visual
comparison of fraction
models to finding patterns for
comparing them when just the
notation is given. Through
their practical work, students
will encounter equivalence.
This includes equivalence to 1.
Students make and play a
fraction strip game and then
record the operations of the
game.
3.44
Content
Number (cont.)
Page 13 of 13
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students can use materials
and diagrams to solve simple
word problems involving
common fractions.
Manipulatives provide the
conceptual framework for
student understanding. By
recording their work, using
fractional notation, students
begin to notice patterns in the
numbers that will enable them
to work symbolically.
Students will:
subtraction of fractions
with the same denominator
How can we add and subtract
fractions?
Adding and subtracting
fractions
Students add fractions using
their fraction strip kit. They
find ways of recording their
equations.
Students are asked if the
following equations are true:
¼+¼+¼=¾
6
/6 – 1 /6 – 1 /6 – 1 /6 = 2 /6
They can explain why or why
not.
fractions: numerator,
denominator, equivalence
How do mathematicians write
fractions?
What is a numerator?
Using vocabulary and
symbols of fractions
Students explore fractions in
a practical nature and use
appropriate vocabulary to
describe what they have done.
Students can use correct
language to describe fractions
and can use symbols to read
and write fractions.
What is a denominator?
• understand and model the
concept of equivalence to
1: two halves = 1,
three thirds = 1
What is equivalence?
Can you show fractions
equivalent to 1?
What patterns do you see in
equivalence to 1?
• reasonably estimate
answers: rounding and
approximation
What are good ways of
making a reasonable estimate
for computation problems?
Why is place value useful in
making estimates? causation
Why do we sometimes round
up or down? causation
How does place value help us
with rounding? connection
Is your answer reasonable?
responsibility
solving a problem.
What is the best way to solve
this equation?
answer need to be?
to solve that problem?
What problem could that
equation help us to solve?
Equivalence
Use fraction kits and
Cuisenaire rods for exploring
fractions and their equivalence
to 1.
Students can explain what
equivalence means.
They can identify equivalence
to 1 using manipulatives.
Students use practical
activities such as cutting fruit,
slices of bread or pizza.
Could that be right?
Students estimate the answer
to a problem, check their
answer using a calculator or a
peer and their work, and
describe how they need to
modify their strategy.
Students read thousand
numbers and practise
rounding these to the nearest
ten and the nearest hundred.
Students suggest numbers that
could round to a given number.
Students are asked to evaluate
the reasonableness of a
variety of answers to a given
problem. They can explain
why or why not. Is 38 a
reasonable answer to 94 - 52?
Why or why not?
Students need many
opportunities to practise
estimating. It is helpful to
establish a range of
reasonable estimates.
Students look at store
catalogues and can select
several items that they
estimate would total just
under a given amount. They
then check the actual costs.
Choosing a method
Students are exposed to a
variety of problem-solving
strategies: guess and check,
use simpler numbers, make a
table or chart, act it out.
Students share their
calculation methods with the
class, who can then verify or
question their thinking.
How do you know that your
method is effective? reflection
How do you know that
someone else’s method is
effective (or not)? reflection
3.45
Page 1 of 18
Students will collect, display and interpret data in a variety of ways. They will compare data displays, including how well they communicate information. They will create and manipulate an
electronic database and set up a spreadsheet using simple formulas to create graphs. They will find, describe and explain the range, mode, median and mean in a set of data, use a numerical
probability scale 0–1 or 0%–100%. They will determine the theoretical probability of an event and explain why this might be different from the experimental probability.
Measurement
Students will estimate, measure, label and compare perimeter, area and volume using formal methods and standard units of measurement. They will develop procedures for finding perimeter,
area and volume and recognize the relationship between them. They will use the correct tool for any measurement with accuracy. They will measure and construct angles in degrees using a
protractor. They will know that the accuracy of measurement depends on the situation and the precision of the tools. They will use and construct 12-hour and 24-hour timetables and be able to
determine times worldwide.
Shape and space
Students will use the mathematical vocabulary of 2-D and 3-D shapes and angles. They will classify, sort and label all types of triangle and quadrilateral. They will turn a 2-D net into a 3-D
shape and vice versa. They will find and use scale and ratio to enlarge and reduce shapes. They will use the language and notation of bearing to describe position, and be able to read and plot
coordinates in four quadrants.
Students will understand and use the relationships between the four operations. They will model and explain number patterns and use real-life problems to create a number pattern following a
rule. They will develop, explain and model simple algebraic formulas. They will model exponents as repeated multiplication, and understand and use exponents and roots as inverse functions.
Number
Students will read, write and model numbers to one million and beyond, extending the base 10 system to the millions and thousandths. They will automatically use number facts. They will read,
write, model, compare and order fractions (including improper fractions and mixed numbers), decimals (to any given place), and percentages. They will interchange fractions, decimals and
percentages. They will add and subtract fractions with related denominators, simplify fractions and explore fractions using a calculator. They will add and subtract decimals to the thousandths and
will model multiplication and division of decimals in the context of money. They will find and use ratios; read, write and model addition and subtraction of integers; and use exponential notation.
They will use and describe multiple strategies to create and solve more complex problems, reasonably estimating the answers. They will select and defend the most appropriate and efficient method.
Content
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• design a survey and
systematically collect,
organize and record the
data in displays:
pictograph, bar graph,
circle graph (pie chart),
line graph
What questions do you need to
ask to collect the information
that will answer your inquiry?
What are the most important
pieces of information we need?
How can you organize the
collection of your data?
different displays that will
assist in your choice of display?
Displaying survey results
Opportunities for designing
and implementing surveys can
be found across the
curriculum.
Students can develop a
strategy to get information
about a topic, process that
data and interpret it.
naturally in the classroom, or
as part of the units of inquiry,
present opportunities for
students to develop surveys
and process data.
As well as graphs, students
should use Venn and Carroll
diagrams to organize and
make sense of data.
It is important to remember
that the chosen format should
without bias.
Why do you need to think
about changing your survey
questions? reflection
• create, interpret, discuss
and compare data displays
(pictograph, pie chart,
bar/line graph) including
how well they
communicate information
Why do you think this data
was displayed in this form?
causation
What did you know already
about this form of graphical
display to choose it for your
data?
The collection and
interpretation of data increases
in complexity as students use
their skill and understanding of
decimal notation, fractions,
percentages and measurement
(scale).
Comparing data displays
Display the same data in a
series of different graphical
displays. Students record and
share their interpretations,
making judgments about the
choice of display and about
which display best represents
the data.
Students complete a decisionmaking organizing chart to
make judgments about the
appropriateness of choosing a
particular display.
From a set of data, students
can choose the most
appropriate display and can
justify their choice.
Students need many
opportunities to see the same
data displayed on a variety of
graphs and charts. They need
to discuss and categorize the
types of information that are
best represented by each type
of graph.
Students can comprehend and
analyse data displayed in a
variety of graphical forms.
Labelling of graphs is
important in the interpretation
of data.
Students can create two
graphical displays for a data
set and can interpret the data.
3.47
Content
(cont.)
Page 2 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
Could you display this data in
a different way?
How would you justify your
choice of display for your
data?
How does the representation
of data change with the use of
different graphical displays?
Why is the information easy
to read and interpret?
Comparing data displays
(cont.)
Students are shown familiar
data in a variety of graphical
displays. They discuss how
the displays affect their
interpretations of the data.
With the same set of data,
students use a computer
software program to generate
different graphical displays.
They discuss the results.
Sentences of students’
interpretations are matched to
a particular graph.
Students can report on how
different graphs clarify or
obscure meaning.
Technology gives us the option
of creating a graph at the press
of a key. Being able to generate
different types of graph, such
as bar graphs, circle graphs
and pictographs, allows
students to explore and
appreciate the attributes of each
type of graph and its efficacy
in displaying the data.
What have we learned that
will help us read and interpret
the data? connection
Can the data be interpreted in
a different way by someone
else? perspective
• find, describe and explain
the range, mode, median
and mean in a set of data
and understand their use
What mathematical skills and
understandings do you need
to apply to determine the
range, mode, median and
mean in a set of data?
How do you identify the
range, mode, median and
mean in this graphical
display?
How can the range, mode,
median and mean be altered
to change the interpretation of
the data?
Why is it important to find
these points of reference for
your data?
How is the median
determined in an even set of
frequencies?
Range, mode, median and
mean
Students investigate a set of
data to find the range, mode,
median and mean.
Given a display of student
attributes, students can answer
the following questions: Who
is the tallest? What is the
difference in height between
the tallest and the smallest
(range)? If a new student
comes into the class, how
would this affect our data?
Students can demonstrate how
the range is determined in a
set of data.
Students work out the
following problem: Ten
students have a mean height
of 125cm; what could be the
height of each student?
possible heights of the students
according to the data given and
can explain their working.
Students calculate the mean
age of all students in the class.
Students can explain how to
find the mean in a given set of
numbers.
Some results are not always
found using whole numbers.
Applying understandings of
decimal notation and of
rounding off can assist with
these results. Practise the
investigation of the mean value
of data using prepared data sets
that result in whole numbers.
Students can order the jumps
according to magnitude and
then identify the range, mode,
median and mean.
Using data already collected
and saved is a simple way to
begin discussions of the
median and to compare it with
the mode. Two further
extensions are to discuss why
the median is sometimes
identical to the mode and why
sometimes it is different, and
why statisticians would
choose to use one method of
calculating the average as
opposed to the other.
Students create a block graph
using a set of interlocking
blocks. They calculate the
mean by taking or adding
blocks to the columns so that
each column becomes the
same height.
Students measure the distance
of a standing jump with paper
ribbon.
Students are given a short
specified time to repeatedly
write a word. Data is collected
and the range is calculated. The
process is repeated for a
calculation activity and the data
is compared and interpreted.
3.48
Content
Page 3 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Working in small groups,
students can set up an
electronic database. They can
reflect on the administration
and setting of tasks; the
collection, organization and
management of data; and the
access of the data.
A database is a collection of
data, where the data can be
displayed in many forms. The
data can be changed at any
time. A spreadsheet is a type
of database where
information is set out in a
table. Using a common set of
data is a good way for
students to start to set up their
own databases. A unit of
inquiry would be an excellent
source of common data for
student practice.
Students can create a
spreadsheet for a set of given
guidelines.
Students will need to be
taught how to use the
software so that it becomes
transparent. This is best done
when there is an authentic
purpose for learning the
software.
Students will:
• create and manipulate an
electronic database for
their own purposes
What does a database look
like? form
What do we need to know in
order to access data from a
prepared database?
How can a database help us to
organize our data?
What are examples of
databases?
Using a database
Students use prepared
databases to extract
information for research
purposes.
Students set up a class
database, setting fields based
on students’ physical
attributes. Students extract
information from the
database in response to a
search query.
How can data be organized
and prepared for display in a
database?
• set up a spreadsheet, using
simple formulas, to
manipulate data and to
create graphs.
How is a spreadsheet similar
to a database? form
How is a spreadsheet unlike a
database?
What is the language of
formulas?
What mathematical
understandings can we apply
to the creation of formulas?
There are ways to find out if
some outcomes are more
“unlikely”, “certain” or
“impossible”. It can be
expressed quantitatively on a
numerical scale.
opportunity to process data
and explore probability in
more thoughtful, efficient and
imaginative ways. The
of technology.
• use a numerical probability
scale of 0 to 1 or 0% to
100%
Will the result be the same if
we repeat the experiment?
Why were these the outcomes
of our game?
How do we make use of
probability in everyday
situations?
Is probability predictable?
What connections do we need
to make between the result of
the outcomes and the use of a
numerical scale?
How do mathematicians
express the probability of an
event?
Setting up a simple
spreadsheet
Students plan and organize a
class picnic. They use a
spreadsheet to calculate the
cost and quantities of items
needed.
Students investigate different
spreadsheets such as the
information found in charts in
a travel brochure.
Students can design and set up
a spreadsheet, enter data and
the formula, or know how to
use the software to enter the
correct formula.
Students use computer
software programs to generate
a spreadsheet.
Probability scales
Students participate in
probability activities such as
tossing coins, and dice games
with one or two dice.
Students draw coloured beads
from a bag.
Students design a game that is
unfair to one or more players.
Students design and use a
random spinning device.
Students determine the
possible outcomes of a chance
experiment.
Using a common set of data
is a good way for students to
start to set up their own
spreadsheets.
Working cooperatively in
groups will assist in the
management of data entry for
larger databases.
This can be assessed orally
with students discussing the
probability of a range of
events.
Students can give a possible
interpretation of a given
probability: “There is a 50%
chance that a green bead will
be pulled from the bag.”
The notion of probability
having a numerical value can
be extended by investigating
the number of possible
outcomes for events.
Students can make statements
of possibility before an event
occurs.
experiences reporting on the
probabilities of events, and
seeing how using the two
different systems clarifies or
obscures meaning. They
should classify the situations
in which they believe
percentages are more useful
and the situations in which
they find fractions more
useful.
Students can describe why
1=100% and can compare a
fraction to a percentage.
This is a good opportunity for
students to explore the use of
the fraction bar as division.
probability of spinning red on
a wheel of four colours.
Why are two systems used?
3.49
Content
(cont.)
Page 4 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Given a situation, students can
determine the theoretical
probability of an event and
can determine the
experimental probability by
collecting and organizing
data. They can compare the
two and formulate theories
about what they notice.
Students need to explore the
differences between
experimental probability and
theoretical probability. They
often have difficulty
accepting the premises of
theoretical probability.
Students will:
• determine the theoretical
probability of an event and
explain why it might differ
from experimental
probability.
Will the result be the same if
we repeat the experiment?
Why/why not?
How do results vary?
Why do results vary?
Are there any patterns?
Theoretical versus
experimental probability
Students frequently repeat
experiments and compare the
results.
Students order events using
available data to answer
questions such as: Which
capital city has a chance of
rain tomorrow?
How is experimental
probability the same as
theoretical probability?
Students understand that, as
more data is gathered,
experimental probability
approaches theoretical
probability.
How is it different?
Why can there be differences?
Technology gives us the
possibility of rapidly
replicating random events.
Computer tools will toss
coins, roll dice, and tabulate
and graph the results.
Given two sets of data,
students can explain why both
sets can be possible results of
the same experiment or
survey.
Measurement
• select and use appropriate
standard units of
measurement when
estimating, describing,
comparing and measuring
How can we accurately
measure a given object?
What units should we use?
What units should we
choose?
A range of thermometers,
jugs, weighing scales etc is
displayed. Students make
statements about what they
see, including observations of
the scale.
Students can take a reading
from a measuring instrument.
Measurements are often
expressed as fractional parts.
Students can find the best buy
of three available sizes of the
same product.
Through their science
activities, students should have
the opportunity to discover the
relationship between grams,
millilitres and cubic
centimetres in order to develop
an understanding of density.
Students can create a flow
chart or other device to
describe how they decide on
the units and tools to use in
different situations.
In real-life measuring
situations, a decision about
which instrument and which
unit to use always precedes
the actual measuring.
Students measure a variety of
objects and record findings on
a chart.
• use measuring tools, with
simple scales, accurately
What do we need to consider
in order to choose an
appropriate instrument of
measurement?
Which instrument is best for
this measurement?
Measuring accurately
Students select from a variety
of measuring tools to complete
a “body measure” of
themselves.
Students draw lines to given
lengths, without the use of
rulers, and solve the problem
of measuring curved lines.
Students draw and construct
objects using accurate
measurements.
• understand that the
accuracy of a measurement
depends on the situation
and the precision of the
tools
How accurate can this
measure be?
How accurate does it need to
be?
What does it depend on?
How accurate is this
measurement?
Students solve problems,
combining objects that must
be kept at a limited weight
and answer questions such as:
What items can be placed in a
suitcase to meet the limit of
20kg? What furniture can I
buy to fit in this room?
Students can write a short
passage using references to
different forms of
measurement (eg a shopping
trip).
Students can use known sizes
of common objects to help
estimate and measure.
Measurement involves
continuous quantities, so
measures cannot be expressed
exactly. The degree of
accuracy depends on the
measuring tool and the
measurer.
Students investigate
weighbridges, costs of
freights and shipping.
cooking activities and follow
a recipe.
Students can determine how
accurate a measurement must
be when cooking, and can
defend their answers.
3.50
Content
Measurement (cont.)
Page 5 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• develop procedures for
finding area, perimeter and
volume
What is area? form
Does everything have area?
form
How can we know the area of
this rectangle without
counting the squares?
How can we calculate the
area of an irregular shape?
Developing procedures for
finding area, perimeter and
volume
Students try to make three
rectangles with an area of
30cm2. They are asked if there
is a limited number of
rectangles they can make and
how they know this.
Students design a garden of a
specific area. They cost the
establishment of the garden
based on its dimensions.
Students can make different
rectangles with a specified
area.
Students can systematically
investigate areas of
rectangles using base 10
materials to lead to the
generalization of length %
width.
What is perimeter? form
Are there short cuts for
finding the perimeter of
polygons?
How can we calculate the
perimeter of an irregular
shape?
What is volume? form
How can we know the volume
of a box without filling it with
cubes?
boxes to determine volume
using cubes. They should
record the procedures they
use with an eye to developing
a standard procedure that can
be generalized and,
ultimately, turned into a
formula.
Students investigate and
compare the volume of objects
using cubes
• determine the relationships
between area, perimeter
and volume
Do all shapes with the same
perimeter have the same
area? (and vice versa)
What can we know about the
perimeter of a rectangle by
knowing its area? connection
What are the relationships
between area, perimeter
and volume?
Students use grid paper or
paper tiles to investigate the
possible arrays of squares for
a given area.
Students can calculate
volumes of 3-D
representations using cubes.
For example, a square-based
pyramid.
Students can make a
rectangular prism with a
volume of 36cm³ and can
describe how they work out the
volume of a rectangular prism.
relationship between area and
perimeter through the use of
grid paper and geoboards.
Students investigate area on
maps and use geoboards to
model area and perimeter.
Why are area, perimeter and
volume like they are?
causation
How are the units of volume
related to the units of length
and area?
Students use isometric dot
paper to draw a
representation of cubes, and
to calculate volume.
Students can explain why
the units of measurement are
related.
making several different
squares and rectangles of a
given area to understand that
the relationship between area
and perimeter is flexible but
connected.
Students generalize their
measuring experiences as they
devise a procedure or formula
for working out area. This
develops algebraic thinking.
Why do the formulas of area,
perimeter and volume work?
causation
How did you discover the
formula?
How do we find the volume of
a box without using cubes?
Why is a cube used to
measure volume?
possible dimensions of the
box by understanding the
procedure of finding volume.
3.51
Content
Measurement (cont.)
Page 6 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
and compare, using formal
methods and standard units
of measurement, the
dimensions of area,
perimeter and volume
What do we need to know
before we can make a
reasonable estimate of
measurement?
How can we test our
estimates?
What do we think about when
we estimate a measurement?
How big is the box?
What are all the possible
dimensions?
Measuring area, perimeter
and volume
Students measure the volume
of quantities using centimetre
cubes and calibrated volume
containers.
Students draw a specified
polygon of specified
perimeters by estimation
(without using a measuring
tool).
Students have 48 balls with a
10cm diameter to pack into
the box.
Students can reflect on their
estimates.
Students can create a 3-D
shape with a volume of
100cm³.
Through their science
activities, students should
have the opportunity to
discover the relationship
between grams, millilitres and
cubic centimetres and should
begin to develop an
understanding of density.
Students can measure the
volume of a box.
How can we convince
someone that our
measurement is correct?
responsibility
• use decimal notation in
measurement: 3.2cm,
1.47kg
How can we deal with the
part that is left over?
How can we apply our
understandings of decimal
notation to measurement
scales?
Using decimal notation
Students are given real objects
and situations, in which they
must make measurements, to
discover that not all
measurements will represent a
whole number.
Students use a calculator to
investigate counting in
decimal sequences.
Students can describe a
situation where they would
need, or want, to record the
measure of an object to the
tenths or hundredths place.
They can explain why using
decimal notation helps them
to understand and describe the
measurement.
Measurement and money
provide the most natural reallife contexts for introducing
students to the need for
decimal notation.
Students solve the following
problem: In a race with five
competitors; the winner had a
time of 12.5 seconds and the
last competitor had a time of
15.2 seconds, what could the
times of the other competitors
be?
• understand that an angle is
a measure of rotation
What is the vocabulary of
measuring rotation?
Angle as a measure of
rotation
Students create rotational
patterns from a stencil and fix
them with a single pivot point.
They rotate the stencil to a
named angle and then trace.
What are real-life examples of
angles?
Students investigate the use of
angles in geometric artworks.
How do you read a circular
scale?
Students draw patterns within
circles.
How do we measure the size
of a turn that we make?
Students can explain
rotational patterns that can
occur in designs.
knowledge of angles can
prove useful in a game of
football or in other games and
sports.
3.52
Content
Measurement (cont.)
Page 7 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
bold.
expectations and
Students will:
• measure and construct
angles in degrees using a
protractor
What language do we use to
describe angles?
How can we identify the
different angles?
between the different angles?
Where are angles to be found
in the immediate environment?
Why are angles a useful form
of measurement?
How do we use a protractor to
create or measure a given angle?
How can we calculate angles
without the use of a protractor?
Measuring and constructing
angles
Students fold squares of paper.
They describe the shapes and
angles created using the
specific vocabulary of angles.
Students use two straws,
joined at an end pivot point,
to create and draw angles.
They order the size of angles
and discuss.
Students draw a chart of
named angles to see the
unique features of each angle
and also to see the
relationships between angles.
Students can identify angles
in their immediate
environment, discuss the
possible measurements of
these angles and justify their
decisions.
Students can use a protractor
to measure angles
appropriately.
Students can draw a rectangle
and check to see if it is
correct.
between angles and geometric
shapes?
How does the measurement of
angles relate to shapes?
manipulating angles, students
of angles with the more
formal mathematical
vocabulary, and begin to
appreciate the need for this
precision.
Students require a lot of
practice in measuring and
constructing angles and in
using a protractor.
Students measure angles in a
real-life context.
Students should compare and
contrast a protractor and a
ruler.
Model use of the protractor.
How are angles connected to
a circle?
The history of measuring
angles is interesting in that it
relates to the rotation of the
earth around the sun.
architectural drawings, art
forms, bridges and buildings.
Students draw a pathway in a
maze following a set of
instructions that uses angles.
How does the measurement of
shapes relate to symmetry?
• use and construct
timetables (12-hour and 24hour) and time lines
What is a 24-hour clock? form
Why do we use 12-hour and
24-hour times?
How do 12-hour and 24-hour
times relate? change,
connection
How do you manage your
time over extended periods?
responsibility
How can you plan a journey
using a timetable?
What do you need to
understand to make sure you
will be on time at a destination?
responsibility
Why is it important to measure
time accurately? reflection
What time would be one
minute before 15.00 hours?
Using time lines and
timetables
Students brainstorm situations
that use 24-hour time.
Students synchronize times on
a 12-hour and a 24-hour clock
and discuss.
Students use authentic rail,
train and bus timetables to
plan a journey within an
authentic context.
Students play card games,
matching cards with 12-hour
and 24-hour times.
Students investigate the
history and use of 24-hour
time.
Students calculate the time
elapsed between two given
times.
Students solve problems using
authentic timetables (eg a TV
programme).
Students can complete a class
timetable on a blank form to a
set of given criteria.
Students can assist with the
organization of a camp
timetable.
Students can create a time line
from midnight to midnight.
The method of counting on
and counting back, using a
time line, is a good way of
seeing and calculating the
time elapsed.
Students may develop
individual strategies for
calculating the time elapsed.
Students can share the
strategies used to convert 12hour time to 24-hour time and
vice versa.
Students can answer
questions such as: The time is
now 25 minutes to 4; how
many different ways can this
be written? A swimming
lesson takes 1 hour and 10
minutes; what could be the
start and finish times of the
lesson?
Students use analogue clocks
to count on or count back to
calculate elapsed time.
3.53
Content
Measurement (cont.)
Page 8 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
All of the sample questions can be
linked to a key concept. Some
examples are noted below in bold.
When given a time in one
country, students can calculate
the time in another country.
This aspect of measurement is
best taught in real-life
contexts.
Students will:
• determine times worldwide.
What are time zones?
Why are there time zones?
causation
What time is it now in your
home country? connection
Times around the world
Students use atlases and globes
to investigate time zones.
Students participate in video
conferencing link-ups with
other schools.
Students interpret flight
schedules.
Shape and space
• use the geometric
vocabulary of 2-D and 3-D
shapes: parallel, edge, vertex
What features do 2-D shapes
have in common?
Using the vocabulary of 2-D
and 3-D shapes
Shape Show and Tell.
Students select a 3-D shape,
list all the possible properties
of the shape and share this
information with others.
What features do 3-D shapes
have in common?
Students create shapes using
straws.
What is an edge?
Students use geoblocks and
similar materials as resources
for identifying the critical
attributes of the shapes.
What language is used to
describe the features of shapes?
What is a vertex?
What is a face?
Students can investigate the
occurrence of flying to a
country on one day and
landing there on the same day.
From a list of 3-D shapes
students can list what real-life
shapes they could be (eg a
sphere and a ball).
Students can write a
procedural text explaining
how to draw a particular
shape.
Students can use the terms
vertex, edge and face to
describe the attributes of 3-D
shapes found in their
immediate environment.
How are the features of 3-D
shapes related to one another?
• classify, sort and label all
types of triangles and
quadrilaterals: scalene,
isosceles, equilateral, rightangled, rhombus,
trapezium, parallelogram,
kite, square, rectangle
What do all triangles have in
common? function
What do all quadrilaterals
have in common? function
Can you change one
quadrilateral into another?
change
How could you describe a
quadrilateral to someone who
has never seen one? reflection
How does the name relate to
the shape?
Sorting triangles
Students explore the features
of these shapes to develop
their own conceptual
framework.
Students can identify and sort
a wide selection of special
triangles and quadrilaterals
and can categorize them in at
least two ways.
this specialized vocabulary,
and the relationship between
the features of a shape and its
name.
Students can identify the types
of angles in their immediate
environment, using the correct
vocabulary, and can defend
their observations.
manipulating angles, students
with the more formal
mathematical vocabulary, and
begin to appreciate the need
for this precision.
Students create a class poster
of shapes and their
descriptions.
Students match a series of
shapes to the text descriptions.
Students draw a series of foursided shapes. They discuss the
They do the same for
triangles.
How does the vocabulary used
to describe a shape relate to
the features of that shape?
vocabulary of types of
angle: obtuse, acute,
straight, reflex
What words do mathematicians
use to describe the different
types of angles?
Why are there distinctions
among these types of angles?
Using the vocabulary of
angles
Students begin to identify and
classify different types of
angles through measurement
activities and exploring
shapes.
How are these angles
connected to one another?
Where can we see examples of
these shapes in the immediate
environment?
3.54
Content
Page 9 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
and transdisciplinary skills (Fig
14 Making the PYP happen).
Students will:
• understand and use
geometric vocabulary for
circles: diameter, radius
circumference
How can you make or model a
circle?
How are all circles the same?
function
Using the vocabulary of
circles
Students investigate drawing
circles using a pencil or chalk
tied to a piece of string. This
will lead to using a pair of
compasses to draw circles.
What makes a circle unique?
• use a pair of compasses
How is a pair of compasses
related to mathematics?
What can be drawn with a pair
of compasses?
Using a pair of compasses
Students follow a set of
instructions to create a pattern
that is based on a line or a
shape.
contrast two circles using the
vocabulary of circles.
Dynamic geometry
programmes are an excellent
resource for student
investigations.
Students can draw a circle
with a diameter of 14cm or a
radius of 8cm.
Students can use a pair of
compasses to investigate how a
particular shape may have been
constructed.
Students can give directions on
how to produce a particular shape.
Using the vocabulary of lines
vocabulary of lines, rays
and segments: parallel,
perpendicular
What are lines, rays and
segments?
Students draw sets of two lines
and discuss them.
How are they related to each
other?
Students find parallel lines in
the classroom. They use a
spirit level to identify
horizontal and vertical lines in
the classroom and then the
parallel lines.
relationship between lines?
When are two lines parallel?
What is a feature of parallel
lines?
Students identify sets of
parallel lines in given
quadrilaterals.
Where do you find parallel
lines?
Students identify lines of
symmetry in shapes.
How do the names of some
shapes reflect the lines used to
draw the shape?
• describe, classify and model
3-D shapes
Why are 3-D shapes named as
they are?
What are the similarities and
differences in 3-D shapes?
Exploring the features of 3-D
shapes
Students explore the features
of shapes to develop their own
conceptual framework.
What clues do the names of
shapes give us to recognize
and draw the shape?
Students create a class poster
of shapes and their
descriptions. They match a
series of shapes to the text
descriptions.
Where does the term
polyhedra originate?
Can you sketch 3-D shapes as
2-D shapes? change
Students use perspective to
draw 3-D shapes.
in the cross-sections of these
3-D shapes? change
Students cut cross-sections of
clay cones, cylinders, pyramids
and prisms. They measure their
dimensions and look for
patterns.
artists draw 3-D shapes.
Students can construct a line
that is perpendicular or
parallel to another.
Students can draw a straight
and a curved set of parallel
lines and can explain how they
constructed them.
Students can identify and mark
all parallel lines as seen in a
geometric drawing.
They can make a design using
only parallel lines.
They can draw a shape of an
object that has 2, 3 or 4 lines of
symmetry.
Students can sort a cube, sphere,
triangular prism and squarebased pyramid into two groups
and justify the reasons for
placing the shapes in these
groups.
The purpose of this study is
for students to understand
specialized vocabulary:
polyhedra, spheres, cylinders,
cones, prisms and pyramids.
Given a photograph of
architectural structures, students
can deconstruct the buildings
into their constituent shapes
using appropriate vocabulary.
Students can predict which
cross-sections come from
which 3-D shapes and can
defend their prediction.
Students can draw a robot
using 3-D shapes.
This is a good opportunity to
conduct an investigation
whose outcomes will lay a
foundation for higher level
algebra and geometry.
Students can locate 3-D/2-D
shapes within artwork.
patterns that occur between
shapes when counting edges,
faces and vertices (Euler).
Leonhard Euler was a
mathematician who was the
first to discover these
relationships.
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Page 10 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
Students will:
• turn a 2-D net into a 3-D
shape and vice versa
What is a net?
How are 2-D shapes related to
3-D shapes?
How is a net created?
What do these shapes have in
common?
Turning 2-D shapes into
3-D shapes and vice versa
Students use interlocking
plastic shapes to create 3-D
shapes.
Students are given sets of
congruent shapes, made from
card, to tape together.
following problem: A set of
six squares was joined
together. They could not be
folded to make a 3-D shape.
What might the pattern of
squares look like?
Students use straws, sticks and
modelling clay to make skeleton
shapes.
Students can design a net for a
box that has a volume of 150
cubed units.
Students require many varied
activities in handling 3-D
shapes so that they see how
the properties of shapes relate
to the name given to those
shapes.
Students require an
understanding of the number
and shapes of faces of 3-D
shapes before they are able to
construct nets.
Students make a city skyline
from various nets.
Students create a container/box
for a toy.
Students use prepared grid
paper to assist in making nets.
• find and use scale (ratios)
to enlarge and reduce shapes
What is scale?
What is a ratio?
What do we need to understand
to enlarge or reduce a shape?
• use the language and
notation of bearing to
describe position
An overhead projector is used
to model enlargements.
How is scale used in real life?
Students do map work.
How is a ratio interpreted?
Students make a scale model
of a city.
What mathematical processes
are used to calculate and
interpret scale?
Students reduce and enlarge
images and patterns through
the use of prepared grid paper
or allow students to determine
the scale for reduction or
enlargement.
Why do plans and maps use
scale?
Using scale
Students model scale by using
base 10 materials.
How are angles used to
describe a bearing?
What is the language used to
describe position?
What is the notation used to
describe position?
Students can estimate the
scale of selected objects
(toys).
Students can draw a reduced
image of an object (chair) and
can explain how they did this
and what scale means.
The idea of scale is very
important in geography. This
mathematical concept is best
practised through a unit of
inquiry.
Students can interpret maps
using scale.
Students can use rectangular
grids to investigate distortions
in images.
Students can solve a problem
such as: “On my biking holiday
I want to travel between
1000km and 1200km. Which
towns can I visit on my
journey?”
Finding a bearing
Students investigate atlases
and maps, and do mapping
exercises.
Students can describe a journey
undertaken by an explorer such
as Captain Cook.
Students should see the
relationships between a half
turn and 180 degrees.
Students give directions to
other students who are
blindfolded.
Students can design their own
grid drawing with
accompanying instructions.
Students will become aware of
the conventions of north and
south being named first in a
direction eg north-east.
different ways in which
accurate directions are given.
Students can plan and
undertake an orienteering
course.
Students follow a set of
written instructions, using
three digit notation of
bearings, to move themselves
or objects from one position to
another.
Students can write a set of three
directions from a designated
starting point to a designated
end point using bearings. They
can follow each other’s
directions.
Navigational systems are
based on bearings.
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Content
Page 11 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
Students will:
• read and plot coordinates in
four quadrants.
In what circumstances can we
use coordinates to describe
position or location?
Coordinates
mapping activities and games
to locate the positions of given
objects.
Students make shapes using
given coordinates on a
pinboard.
Cartesian coordinates are
introduced as a way of
describing position.
to understand how
of algebra.
(inverse function)
• model and explain number
patterns
multiplication connected?
How is multiplication repeated
addition?
division related? connection
subtraction related? connection
How are numbers arranged to
create a pattern?
between the numbers?
How can you explain what
will be the next number in this
pattern?
How can an array relate to a
function?
Where can number patterns be
found in real-life situations?
The relationship between
Using calculators, students
explore the results of repeated
addition against multiplication
of numbers.
Students can create their own
map and include grid
references.
Cartesian coordinates (grid
points) are introduced as a
way of describing position.
Students can identify positions
on maps from given
coordinates.
Cartesian coordinates use a
pair of axes at right angles to
each other. Students need to
have an understanding of
positive and negative
numbers. Positions are located
on the intersections of the
perpendicular lines from each
axis. The first coordinate is
the reference point from the
horizontal axis and the second
coordinate is the reference
point from the vertical axis.
Students can list coordinates
for given positions.
Students can plot a point on a
grid according to certain
specifications.
Students can use and explain
addition and multiplication.
Using tiles, students create
rectangular arrays that have
equal areas but different
dimensions.
possible dimensions of a
garden when its area is 24
square units.
Using manipulatives and
calculators, students explore
this relationship.
Students can use and explain
division and subtraction.
Explaining number patterns
patterns in multiplication
tables.
Students use a circle, with
numerals 0–9 placed evenly
around the circumference, to
create shapes by drawing to
the number that forms the last
digit of the product: a 10 point
star is created when
investigating 3s.
Students identify and explain
products from a multiplication
table on a 100s chart.
Students draw “Input Output
Machines”, each with an
identified function, and answer
questions such as: What will
be the output if the input is 4?
Students can continue a pattern
by using plastic counters.
Students can continue patterns
by making a series of congruent
shapes on grid paper.
Students can create a pattern
from a given starting point.
Students can identify a pattern
and explain the pattern or rule
in a series of related numbers:
39
93
%26
%62
Students can follow and
determine a rule.
Students can use a calculator
to complete functions.
Fibonnacci sequences.
3.57
Content
Page 12 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
Students, given a real-work
situation, can write a function
that describes it.
Generalizing a pattern and
calling it a function is a central
idea in mathematics. Students
need practice in the formation
of functions, and in
determining functions from a
t-chart (graph), in order to
develop algebraic thinking.
Students will:
• use real-life problems to
create a number pattern,
following a rule
How can functions help us to
describe real-life problems?
connection
What patterns and
relationships are there?
Creating patterns that
follow a rule
Using a t-chart (graph),
students record the findings of
a real-life situation such as: If
parents double their allowance,
how much money will they
have at the end of one week,
two weeks, three weeks.
Students make up a four digit
number and give reasons as to
why it is easy to remember.
• develop, explain and model
simple algebraic formulas
in more complex equations:
x + 1 = y, where y is any
even whole number
What symbols do
mathematicians use to
represent numbers that are
unknown?
How do we determine a
missing quantity?
Cracking codes
Students use a series of
symbols to write different
related equations: What do the
symbols represent?
Missing number activities.
These can be in horizontal and
vertical notation.
such as: “My mother is double
my age; how old might I be?”
Students can make a house
using six matchsticks.
Students can verbalize their
thinking and understanding of
number relationships:
100 ÷ 4 = 20. What needs to
be changed? Explain your
strategies for making the
statement true.
Algebra is a mathematical
language using numbers and
symbols to express
relationships. When the same
relationship works with any
number, algebra uses letters to
represent the generalization.
Letters can be used to
represent the quantity.
Students will need to
understand that an equation
has a balance: 2 + x = 5
Students can write a sentence
that describes a real-world
situation where using a
variable is useful.
Students progress to
understanding that some
algebraic formulas have a
range of solutions.
Students can use concrete
materials to model and explain
square numbers.
A prerequisite to the use of
exponents is the understanding
of multiplication and the
notion of factor. An exponent
tells us how many times the
number is a factor.
What relationships can we see
between algebraic formulas
and the four processes?
Students practise using
variables throughout the year
when constructing formulas
for observed mathematical
patterns.
• model exponents as
repeated multiplication
What is an exponent?
How does an exponent relate
to a given digit?
How does an exponent relate
to repeated multiplication?
Exponents as repeated
multiplication
Students investigate square
numbers by drawing patterns
of dots in squares:
1%1 2%2 3%3 4%4
They relate this to the use of
an exponent.
Students investigate cubed
numbers:
1 % 1 % 1, 2 % 2 % 2
understand and use exponents
and roots as inverse functions:
9², √81.
division connected?
How do exponents and roots
relate to each other?
Using exponents and roots
Students examine a variety of
models of squares on grid
paper. They compare and
contrast the measure of the
sides and the areas. This data
can be represented in t-charts
(graphs) and formulas.
Students can use concrete
materials to model and explain
cubed numbers.
Students can explain how x
squared is related to the square
root of x, and how they know
this.
3.58
Content
Number
on its place within a base 10
The degree of precision
needed in calculating depends
on how the result will be used.
opportunity to explore
relationships and rules in the
number system. Students now
have the opportunity to use
technology to find patterns,
explore relationships and
develop algorithms that are
meaningful to them. The
of technology.
Page 13 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
Students will:
system, to millions and
beyond; and to thousandths
and beyond
What is place value?
How does our knowledge of
place value relate to how we
read large numbers?
between a place value system
and the number of digits it
uses?
How does a value change in a
place value system?
How are the different places in
a place value system related to
one another? connection
Using numbers to a million
Making different numbers
from a set of numbers rolled
on a die.
Students discuss and look for
patterns in the way large
numbers are verbalized.
Students use number
expanders and cards to model
large numbers.
Students make large numbers
in dice and card games. They
make the highest/lowest
number.
Students record how numbers
are part of their lives
(eg telephone numbers).
Use two sets of 0–9 cards.
Draw out seven cards. Make
the highest and lowest
number. Ask six students to
write their number on a card
and see whether they can
place these in order from the
largest to the smallest. Allow
students to discuss the
strategies they used.
Students can model large
numbers on an abacus. They
can discuss the value of each
digit and of zero.
It is not feasible to continue to
develop and use base 10
materials beyond 1000.
Students should have little
difficulty in extending the
place value system once they
have understood the grouping
pattern up to 1000.
The naming of the places
follows a simple and selfexplanatory pattern.
Use base 10 materials online
at
http://www.edbydesign.com
Students can make a poster of
large numbers found in
newspapers.
Students know which digit has
the greatest value in the
number 22 022. They can
explain how they know this.
How does the base 10 system
extend to millions and
beyond?
What rules of place value do
we apply to create larger
numbers?
the following scenario: a new
student joins their class and
has missed this unit on place
value. Can you explain to
them how the base 10 system
works.
Student can explain how the
base 10 system works to
another student and can
demonstrate this using
manipulatives.
Students present a counting
system to the class.
What is a decimal?
How is a decimal represented?
How are decimals related to
whole numbers?
How are decimals connected
to the number 1?
occurrence of a decimal point
in a number.
Discuss with students what a
decimal place represents.
Money and measurement are
good examples for discussion
of decimal notation.
Students can explain how, in
our place value system, the
places to the right of the
decimal point resemble and
differ from the places to the
left.
They can explain what a
decimal fraction is.
Students use base 10 material.
The 1000 block now
represents one unit.
They can explain how decimal
fractions are changed by
multiplying by 10/100/1000.
Model numbers using graph
paper.
such as: What numbers can be
in between 2.45 and 3.16?
_._ + _._= _._
Students key a number into a
calculator. They make the
number ten times larger but
predict what will happen before
they key in × 10.
Investigations and discussions
should lead to an understanding
that each place value column
to the left is 10 times greater
and each place value column
to the right is 10 times smaller.
3.59
Content
Number (cont.)
Page 14 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
Students can match cards with
an example of an algorithm
and the worded strategy.
To be useful, the number facts
need to be recalled
automatically. Teaching
strategies for obtaining the
facts will make learning more
authentic and easier. A
thorough understanding of the
operation and the use of
efficient strategies is essential.
Drill and practice is required
for immediate recall of facts.
Students can explain the steps
in multiplying 45 × 23.
The introduction of a
remainder must only follow
after students have a strong
understanding of the division
algorithm. Later, decimal
notation can be introduced
into the answer.
Students will:
• automatically recall and use
basic number facts
memorize the multiplication
facts?
memorize the division facts?
What patterns can you see in
multiplication tables?
How do these patterns assist
in your recall of facts?
What strategies can you use to
add and subtract mentally?
Number facts
Students are exposed to a
variety of activities and
strategies to enable them to
construct meaning about basic
number facts. They explore
several methods and discuss
these with other students.
Basic multiplication facts
• using arrays
• known facts: 5s 10s 2s
(doubling). Investigate the
patterns made in
multiplication tables.
• 100s chart
• geometric shape created on
a circle marked at intervals
from 0–9
• place value: using 10s, 100s
etc to check reasonableness
of answers
• “Serial” number facts:
6 % 7 = 42, 60 % 7 = 420.
Students comment on the
patterns they see.
Basic division facts
• arrays to show sharing
• think of the multiplication
fact
• find the missing factor.
• create and solve multiple
digit multiplication and
division problems
How does place value relate to
our understanding of
multiplying by two or more
digits? function
problems
multiplication by 10 and
develop a rule. Units become
tens and there are no ones.
Students combine
understandings of multiplying
by 10 and multiplying by a
single digit. They check
answers by rounding off and
approximating an answer.
addition and subtraction of
fractions with related
denominators
What is a fraction?
How does a fraction relate to a
whole number?
How is a fraction represented?
How can we add and subtract
fractions of different sizes?
Addition and subtraction of
fractions
Students are provided with
various materials to explore
fractions. They use these to:
• make models of fractions
• create the same fractions
from different materials
• identify how many parts
make a whole
• compare fractions
• discover equivalent fractions.
Students use fraction strips
and fraction dice to add and
subtract fractions.
Students can make up word
problems.
Students can solve problems
such as: Use 9, 8, 7 and use %.
How many different answers
can you get?
‫ ٱٱ‬% ‫= ٱ‬
‫ ٱٱ‬% ‫= ٱٱ‬
‫ ٱٱٱ‬% ‫= ٱ‬
Students can explain whether
the following equations are
true or false:
¼ + ⅛ + ⅛ + ½ = 1,
1 – ⅛ – ¼ - ½ = 0.
Related denominators are
those such as halves, quarters
and eighths. Through
equivalence, students can
substitute so that all
denominators are the same.
A fraction is part of a whole.
Whole numbers can also be
expressed in the same way.
Explore fractions at
www.math.rice.edu.
3.60
Content
Number (cont.)
Page 15 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
demonstrate their understanding in
a variety of ways.
Students will:
improper fractions and
mixed numbers
What is an improper fraction?
What is a mixed number?
How are improper fractions
and mixed numbers
connected? connection
Improper and mixed
fractions
Colour a Fraction game:
Students roll a fraction dice
and colour the parts of a
shape. Extend this game so
that answer sheets may have
more than one line showing
halves. It may be possible to
colour in 6 halves.
Pizza game: Students roll a
dice to make up pizzas from
fractional slices.
Students can use fraction kits
to model fractions. They can
demonstrate making of
“wholes” using fractional kits
and explain using
mathematical language.
The understanding that a
denominator is the way to
record the number of equal
parts of a whole is needed
before the concept of
improper fractions can be
understood.
Students can use Cuisenaire
rods to show the relationships
and understandings of
improper fractions.
Students fill labelled bottles
by using ½ cup measures.
Students use fractional
number lines to see the
relationship between improper
fractions and mixed numbers.
• compare and order fractions How do we know that a
fraction is smaller/bigger than
another?
How can two fractions be
compared?
How do we compare two
fractions with different
denominators?
• model equivalency of
fractions: 2/4 = ½
Why are these two fractions
the same?
What patterns do you see in
equivalent fractions?
• simplify fractions
Why do we simplify fractions?
What mathematical
understandings do we use to
simplify fractions?
• use the mathematical
vocabulary of fractions:
improper, mixed numbers
What is the language of
fractions?
How is the language of
fractions connected to other
mathematical language?
Comparing and ordering
fractions
Students use number lines to
show the order of fractional
parts.
Students use a number line
with marked fractional
intervals to count in fractional
steps.
Students cut two one-metre
pieces of string at ¾ and 7/8 of
its length. Students can
investigate and explain
strategies for discovering
which is the longest.
Students use fractional strips
to compare fractions.
Equivalency of fractions
Students use fraction kits, and
match 3/6 with ½.
Students can demonstrate why
two fractions are equivalent.
sandwiches can be cut up,
using pieces to investigate
equivalence.
Record a series of equivalent
fraction pairs. Students can
explain why each pair is the
same.
Lose a Whole game: Place
fractional parts along a whole
strip. Roll the dice and take
away the fraction rolled.
Equivalent fractions can be
used.
Two fractions added together
give a total of ½. Students can
explain what these two
fractions might be.
Simplifying fractions
Students investigate number
factors to find common
denominators.
Students can show fractions in
their simplest form.
Using vocabulary of
fractions
Write a list of fractions: 2/5, ¾,
½, 11/3. Ask students how
these can be related and into
which groups they can be
placed.
Students can use the correct
mathematical language and
notation to describe fractions.
Multiplying both the
numerator and denominator
successively by 2, 3, 4, 5 will
lead to a generalization about
renaming fractions.
The best way to name a
fraction is usually in its
simplest form. Reduce a
fraction to its simplest form by
dividing both the numerator
and denominator by their
highest common factor.
They can group fractions and
describe them accurately.
3.61
Content
Number (cont.)
Page 16 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
a variety of ways.
Students consider the
following equation 0.3 + 0.5 =
0.7 and can answer the
following questions: Is this
equation true? Why or why
not? How is this problem the
same as, and different from, 3
+ 5?
Manipulatives provide the
conceptual framework for
understanding. By recording
their work using decimals,
students begin to notice
patterns that will enable them
to ultimately work
symbolically.
They consider the equation 2.9
– 1.4 = 1.5. Is this equation
true? Why or why not? How is
this problem the same as and
different from 29 – 14?
The development of addition
and subtraction of whole
numbers and decimals follows
the consolidation of
understanding and recording
of tenths.
Students will:
• read, write and model the
addition and subtraction of
decimals to the thousandths
What is the connection
between fractions and
decimals?
How is a decimal a fraction?
How does addition and
subtraction work with
decimals?
How is this connected to what
you know about place value?
Addition and subtraction of
decimals
Base 10 materials are used to
model simple addition and
subtraction.
Model using squares divided
into tenths and hundredths.
Using measurement (eg 1.6m)
as an example, students can
answer questions such as:
How much more is needed to
make 2.0m into 2.7m?
Students join three lengths of
string together and measure to
find the total length. They
confirm the total by using a
different method. Students
share the different methods.
of decimals (with reference
to money)
• round decimals to a given
place or whole number
What does the decimal point
represent in money terms?
What happens to the values
when they are
multiplied/divided by
multiples of 10?
Why do we want to round to
decimal places?
When do we need to be less
precise/more precise?
of decimals (using money)
Shopping activities. Students
work with shopping receipts.
percentages
To what do percentages
relate?
What are real-life examples of
percentages?
Why are percentages used in
mathematics?
Students can apply their
understanding of decimal
notation to calculating money.
Estimation plays a key role in
checking the feasibility of
answers. The method of
multiplying numbers and
ignoring the decimal point,
then adjusting the answer by
counting decimal places, does
not give the student an
understanding of why it is
done. Application of place
value knowledge must
precede this application of
pattern.
Students can round to a given
decimal place by using their
knowledge of rounding to ten.
Situations in everyday life
often demand approximate
answers or working to
hundredths.
Students calculate sports
scores and follow the
procedure for calculating
diving or gymnastics scores.
Rounding decimals
Shopping activities involving
multiple purchases. Students
calculate the cost per item and
must round their purchases to
the nearest cent, penny etc.
Two decimal numbers of three
digits are added together then
rounded off to 3.3. Students
can explain what the numbers
could have been.
What is a percentage?
When told that the difference in
height of the tallest and shortest
person in the class is 0.35m,
students can find the heights of
these two people.
Percentages
Use base 10 materials to
model percentages.
percentages in newspapers
and magazines and interpret
what they mean.
Students find and read
percentages used in sale
tickets.
Students can explain what the
missing numbers could be
when an algorithm is set up
with missing digits in the
addends and total.
Students can model
percentages using base 10
materials and can explain their
ideas.
They can read and write
percentages they find in
newspapers and magazines and
begin to interpret occurrences
of percentages.
3.62
Content
Number (cont.)
Page 18 of 18
learn?
Notes for teachers
Sample questions
Sample activities
Sample assessments
a variety of ways.
From a variety of possible
strategies, students can select
the most efficient method and
explain how they reached their
conclusion.
The research is unequivocal:
people invent or adopt
methods that make sense to
them. The methods they use
are varied, flexible and often
lead to good estimation skills.
Calculator skills must not be
ignored. All answers should
be checked for their
reasonableness.
Students will:
• select and defend the most
appropriate and efficient
method of solving a
problem: mental estimation,
mental arithmetic, pencil
and paper algorithm,
calculator.
How is mathematics a
language?
How do we interpret
mathematical language?
What cues are in language to
help us select the operation
needed to solve a problem?
How do we know when to use
a particular process?
What words are signposts to
choosing an operation to solve
a problem?
Choosing the best method
to solve problems
written and numerical
problems to solve. They have
the opportunity to change their
methods, discuss their changes
and to explain their choices.
Students can estimate an
answer to a problem, check it
against a calculator and then
explain why or why not they
should/should not modify their
strategy.
When rounding off, the use of
a card to cover the digits to be
rounded off allows students to
see the two components of the
number more clearly.
When do we use estimation in
mathematics?
3.64

Mathematics - Drs International School

Transcription

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